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<h2 class="hd hd-2 unit-title">7.1. Piecewise-continuous functions.</h2>
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<h3 class="hd hd-3 problem-header">Objectives.</h3><p>
After completing this lecture the student will be able to </p><ol class="enumerate"><li value="1"><p>
use the <b class="bfseries"><span style="color:#0000FF">unit step function</span></b> to give formulas for discontinuous functions. </p></li><li value="2"><p>
recognize <b class="bfseries"><span style="color:#0000FF">left</span></b> and <b class="bfseries"><span style="color:#0000FF">right values</span></b> and <b class="bfseries"><span style="color:#0000FF">piecewise-continuous</span></b> functions. </p></li><li value="3"><p>
employ the <b class="bfseries"><span style="color:#0000FF">[mathjaxinline]t[/mathjaxinline]-shift rule</span></b> to Laplace transforms and their inverses. </p></li><li value="4"><p>
describe and compute the <b class="bfseries"><span style="color:#0000FF">step response</span></b> of an LTI system using Laplace transform. </p></li></ol>
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<h2 class="hd hd-2 unit-title">7.2. Discontinuous functions.</h2>
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<h3 class="hd hd-2">Unit step functions</h3>
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<p>
Discontinuous functions provide important tools in representing nature. They become useful whenever there are two time scales at play. </p><p>
For example, I turn on an electric light. The brightness goes from 0 to a large positive value apparently instantaneously. This is nicely modeled by the <b class="bfseries"><span style="color:#0000FF">Heaviside</span></b> or <b class="bfseries"><span style="color:#0000FF">unit step function</span></b> </p><table id="a0000000727" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t)=\begin{cases} 0& \quad t<0\\ 1& \quad t>0\end{cases}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><img src="/assets/courseware/v1/13af9bd8851fe2b7d295fa20401899b8/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside.svg" width="200px" style="margin: 0px 10px 10px 10px"/></center><p>
We leave the value of [mathjaxinline]u(t)[/mathjaxinline] at [mathjaxinline]t=0[/mathjaxinline] <i class="itshape">undefined</i>. </p><p>
If we were to look at this process at a finer time scale – milliseconds rather than seconds – we would see a graph like this: </p><center><img src="/assets/courseware/v1/20adbd95f3ec846aa126359805ab3ec7/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_zoom.svg" width="220px" style="margin: 0px 10px 10px 10px"/></center><p>
Or there might even be overshoot, like this: </p><center><img src="/assets/courseware/v1/9f6d42735decfb037f294c872aaf8b66/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_overshoot.svg" width="220px" style="margin: 0px 10px 10px 10px"/></center><p>
The discontinuous function [mathjaxinline]u(t)[/mathjaxinline] is an idealization of an event happening too fast for us to see, or to care about. </p><p>
The unit step function can be used to build many other useful discontinuous functions. </p><ol class="enumerate"><li value="1"><p>
We can represent the light turning on at some later time [mathjaxinline]t=a[/mathjaxinline] by shifting the unit step function: [mathjaxinline]u(t-a)[/mathjaxinline] </p><center><img src="/assets/courseware/v1/948376106647f59d251bebcac4878132/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_shift.svg" width="200px" style="margin: 0px 10px 10px 10px"/></center></li><li value="2"><p>
Our functions will usually be of interest only for [mathjaxinline]t>0[/mathjaxinline]; so it is often useful to write [mathjaxinline]u(t)f(t)[/mathjaxinline] to set the values of the function to zero for [mathjaxinline]t<0[/mathjaxinline]. Here's the graph of [mathjaxinline]u(t)\cos (t)[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/883eb769e4de5d86bbc6de4a9afc0f55/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_cos.svg" width="500px" style="margin: 0px 10px 10px 10px"/></center></li><li value="3"><p>
We may want to shift a function to the left or to the right. To make sure we are still only considering [mathjaxinline]t>0[/mathjaxinline], we might shift right by [mathjaxinline]a[/mathjaxinline] units and then clip, like this: [mathjaxinline]u(t)f(t-a)[/mathjaxinline]. Here's the graph of [mathjaxinline]u(t)\cos (t-\pi /6)[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/b0268541384a446b8a34fc91835e9f3c/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_cos_shiftA.svg" width="600px" style="margin: 0px 10px 10px 10px"/></center><p>
If we want to leave zeros behind after shifting to the right by [mathjaxinline]a[/mathjaxinline] units, we can use [mathjaxinline]u(t-a)f(t-a)[/mathjaxinline]. Here's the graph of [mathjaxinline]u(t-\pi /2)\cos (t-\pi /2)[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/47bfdfd8db82f00f5befdb6e267a01fc/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_cos_shift.svg" width="600px" style="margin: 0px 10px 10px 10px"/></center></li><li value="4"><p>
The [mathjaxinline]a,b[/mathjaxinline]-<b class="bf">window</b> has the graph. </p><center><img src="/assets/courseware/v1/97d6914b2dfad643a7200b2d0c91e3af/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_window.svg" width="300px" style="margin: 0px 10px 10px 10px"/></center><p>
It's given by the formula </p><table id="a0000000728" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)-u(t-b)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In thinking about functions built from step functions, read them from left to right: the window turns on at time [mathjaxinline]a[/mathjaxinline], and then turns off at time [mathjaxinline]b[/mathjaxinline]. </p></li><li value="5"><p>
We can “clip" a part of some other function by multiplying it by the window: [mathjaxinline](u(t)-u(t-\pi ))\cos t[/mathjaxinline] has graph </p><center><img src="/assets/courseware/v1/e66f985abbd9f067c66dc18af4647c60/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_heaviside_window_cos.svg" width="300px" style="margin: 0px 10px 10px 10px"/></center><div><br/></div>The window turns on at [mathjaxinline]0[/mathjaxinline], and turns off at [mathjaxinline]\pi[/mathjaxinline]. Thus this function looks like [mathjaxinline]\cos t[/mathjaxinline] on the interval from [mathjaxinline]0[/mathjaxinline] to [mathjaxinline]\pi[/mathjaxinline], and is zero outside of this interval. </li></ol>
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Find the formula for the function [mathjaxinline]f(t)[/mathjaxinline] with the following graph. </p>
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<text> [mathjaxinline](t-1)\left(u(t-1)-u(t-2)\right)[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">7.3. Piecewise continuity.</h2>
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It's good to be precise about what kind of functions we are willing to consider in our study of systems and signals. While we want to accept certain discontinuities, we don't want functions like [mathjaxinline]f(t)=1/t[/mathjaxinline] that blow up to infinity in finite time. In order to control what happens near discontinuities, we recall the definition of the one-sided values of a function: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:right; border:none">
The <b class="bfseries"><span style="color:#0000FF">left value</span></b> is </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle f(a^{-})=\lim _{t \rightarrow a^-}f(t)=\lim _{t \uparrow a}f(t)[/mathjaxinline] . </td></tr><tr><td style="text-align:right; border:none">
The <b class="bfseries"><span style="color:#0000FF">right value</span></b> is </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle f(a^{+})=\lim _{t \rightarrow a^+}f(t)=\lim _{t \downarrow a}f(t)[/mathjaxinline] . </td></tr></table><p>
(Here [mathjaxinline]t\uparrow a[/mathjaxinline] denotes the limit from below, and [mathjaxinline]t\downarrow a[/mathjaxinline] denotes the limit from above). </p><p><p><b class="bfseries">Example 3.1 </b> Recall the <b class="bfseries"><span style="color:#0000FF">Heaviside</span></b> or <b class="bfseries"><span style="color:#0000FF">unit step function</span></b> </p><table id="a0000000730" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t)=\begin{cases} 0& \quad t<0\\ 1& \quad t>0\end{cases}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
For the unit step function, [mathjaxinline]u(0^{-})=0[/mathjaxinline] while [mathjaxinline]u(0^{+})=1[/mathjaxinline]. We don't really care about the value of [mathjaxinline]u(t)[/mathjaxinline] at [mathjaxinline]t=0[/mathjaxinline], and have left it undefined. The Heaviside function is an example of a <b class="bf"> piecewise continuous</b> function. </p></p><p><b class="bf">Definition.</b> A function [mathjaxinline]f(t)[/mathjaxinline] is a <b class="bfseries"><span style="color:#0000FF">piecewise continuous function</span></b> if it is defined and continuous except at a discrete collection of points, but at each of them both left and right limits exist. </p><p>
Here is a graph of a piecewise continuous function. </p><center><img src="/assets/courseware/v1/d5ad4f7754f53092d8ca02ee5f8d07d9/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_discontinuous.svg" width="300px" style="margin: 0px 10px 10px 10px"/></center><p>
Notice that a piecewise continuous function is allowed to be undefined at some points. Of course, if [mathjaxinline]f(t)[/mathjaxinline] is not defined at [mathjaxinline]t=a[/mathjaxinline] then we require that the left and right limits exist at [mathjaxinline]a[/mathjaxinline]. If [mathjaxinline]f[/mathjaxinline] is continuous at [mathjaxinline]a[/mathjaxinline] then the left and right limits are the same and both equal [mathjaxinline]f(a)[/mathjaxinline]. So, in a sense we <b class="bf">never</b> care about the value of [mathjaxinline]f(t)[/mathjaxinline] at any single point. If [mathjaxinline]f(t)[/mathjaxinline] is continuous at [mathjaxinline]t=a[/mathjaxinline] then we can reconstruct the value [mathjaxinline]f(a)[/mathjaxinline] from knowing neighboring values, and if it is not continuous then we can just leave [mathjaxinline]f(a)[/mathjaxinline] undefined. </p><p>
Here's a nice feature of piecewise continuity: If [mathjaxinline]f(t)[/mathjaxinline] is piecewise continuous in the finite interval [mathjaxinline][a,b][/mathjaxinline], then </p><table id="a0000000731" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\int _ a^ b f(t)\, dt[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
exists (and has a finite value). </p><center><img src="/assets/courseware/v1/3350b33e8040f2cf2d32f2c074398568/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_piecewisewitharea.svg" width="300px" style="margin: 0px 10px 10px 10px"/><br/>Area under the piecewise continuous curve is well-defined and finite.<br/>Negative areas depicted as orange shaded regions.<br/>Positive areas depicted as blue shaded regions. </center>
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<h2 class="hd hd-2 unit-title">7.4. Laplace transform of unit step function.</h2>
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<h3 class="hd hd-2">On the failure of inverse Laplace transform to be unique</h3>
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<p><b class="bfseries"><span style="color:#FF7800">Note on video:</span></b> Up until now, we said that the inverse Laplace transform of [mathjaxinline]\, 1/s \,[/mathjaxinline] was [mathjaxinline]1[/mathjaxinline]. Because the Laplace transform does not see the function for [mathjaxinline]t<0[/mathjaxinline], the inverse transform is ambiguous. The video attempted to remove this ambiguity by requiring that the inverse Laplace transform to be zero for [mathjaxinline]t<0[/mathjaxinline] — that is, multiply by [mathjaxinline]\, u(t) \,[/mathjaxinline] the unit step function. Now the inverse Laplace transform of [mathjaxinline]\, 1/s\,[/mathjaxinline] is [mathjaxinline]\, u(t)[/mathjaxinline]. </p><p>
Unfortunately, this still leaves some ambiguity. Both because the Laplace transform is defined as an integral and because the integral is over the range [mathjaxinline]t>0[/mathjaxinline] there are several limitations on how completely you can reconstruct [mathjaxinline]\, f(t)\,[/mathjaxinline] from its Laplace transform [mathjaxinline]\, F(s)[/mathjaxinline]. </p><ul class="itemize"><li><p>
You can't say anything about [mathjaxinline]\, f(t)\,[/mathjaxinline] for [mathjaxinline]t<0[/mathjaxinline]. </p></li><li><p>
You can't say anything about [mathjaxinline]\, f(a)\,[/mathjaxinline] for any specific value [mathjaxinline]t=a[/mathjaxinline]. </p></li></ul><p>
As examples, think about the Laplace transform of the three functions with graphs displayed below. </p><center><img src="/assets/courseware/v1/fb0409fefa8b5ad1c992f56e5afc427d/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_steps_nonunique.svg" width="700px" style="margin: 0px 10px 10px 10px"/></center><p>
In each case, there's no problem defining the integrals for the Laplace transform, and all three integrals converge (for [mathjaxinline]\mathrm{Re\, }(s)>0[/mathjaxinline]) and converge to the same values. The Laplace transform of all three functions is [mathjaxinline]1/s[/mathjaxinline] (for [mathjaxinline]\mathrm{Re\, }(s)>0[/mathjaxinline]). </p><p>
If we want to come closer to making the inverse transform well-defined, we can do several things. </p><ol class="enumerate"><li value="1"><p>
Assume [mathjaxinline]f(t)=0[/mathjaxinline] for [mathjaxinline]t<0[/mathjaxinline]. So rather than saying that </p><table id="a0000000732" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\mathcal{L}^{-1}\left(\frac{1}{s^2+1};t\right)=\sin (t),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
we can say </p><table id="a0000000733" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\mathcal{L}^{-1}\left(\frac{1}{s^2+1};t\right)=u(t)\sin (t).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li><li value="2"><p>
While we can't recover [mathjaxinline]f(a)[/mathjaxinline] itself for any specific value of [mathjaxinline]a\geq 0[/mathjaxinline], it turns out that we <b class="bf">can</b> determine both left and right limits at [mathjaxinline]a[/mathjaxinline], </p><table id="a0000000734" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f(a^-)=\lim _{t\uparrow a}f(t)\quad \hbox{and}\quad f(a^+)=\lim _{t\downarrow a}f(t)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
(Not obvious, but true, at least assuming [mathjaxinline]\, f(t)\,[/mathjaxinline] is piecewise continuous.) So, if these two limits coincide, we can specify that our function should be continuous at [mathjaxinline]t=a[/mathjaxinline] and declare the value [mathjaxinline]\, f(a) \,[/mathjaxinline] to be that common value. If the two values differ, the function [mathjaxinline]\, f(t) \,[/mathjaxinline] exhibits a jump discontinuity at [mathjaxinline]t=a[/mathjaxinline], and it's best to leave [mathjaxinline]\, f(a) \,[/mathjaxinline] undefined. </p></li></ol>
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<h2 class="hd hd-2 unit-title">7.5. The t-shift rule.</h2>
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<h3 class="hd hd-2">Motivating the t-shift rule</h3>
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We have made a general agreement that we start observing our system at [mathjaxinline]t=0[/mathjaxinline]. But perhaps the action actually begins a little later! So perhaps the music starts to play not at [mathjaxinline]t=0[/mathjaxinline] but rather at [mathjaxinline]t=a[/mathjaxinline]. </p><p>
There are two ways this might happen. </p><ol class="enumerate"><li value="1"><p>
Perhaps the music was playing all along, but we turned on the amplifier at [mathjaxinline]t=1[/mathjaxinline]. If [mathjaxinline]\sin (\omega t)[/mathjaxinline] represents the sound (pressure) wave at our ear, what we hear is </p><table id="a0000000735" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-1)\sin (\omega t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li><li value="2"><p>
On the other hand, perhaps the band really only began to play at time [mathjaxinline]t=1[/mathjaxinline]. Then what we hear is modeled by </p><table id="a0000000736" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-1)\sin (\omega (t-1))[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></li></ol><p>
In the second scenario, we say that the signal ([mathjaxinline]\sin (\omega t)[/mathjaxinline] in this case) has been <b class="bfseries"><span style="color:#0000FF">delayed</span></b> by 1 second. Generally, </p><table id="a0000000737" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t) \quad \text { is }\, f(t) \, {\color{blue}{\text { restricted to the window}}} \, t\geq a[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><table id="a0000000738" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t-a) \quad \text { is }\, f(t) \, {\color{blue}{\text { delayed }}} \text { by } \, a \, \text { time units}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><img src="/assets/courseware/v1/e19e26ffc7718dd14fba88630c775d18/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_restricted.svg" width="325px" style="margin: 0px 10px 10px 10px"/></td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/6312057380e36b36abfeb355fbaf2836/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_delayed.svg" width="375px" style="margin: 0px 10px 10px 10px"/></td></tr><tr><td style="text-align:left; border:none">
Restricted to [mathjaxinline]t>a[/mathjaxinline] </td><td style="text-align:left; border:none">
Delayed by [mathjaxinline]a[/mathjaxinline] units. </td></tr></table></center><p>
The graph of [mathjaxinline]\, u(t-a)f(t-a) \,[/mathjaxinline] is the graph of [mathjaxinline]\, f(t) \,[/mathjaxinline] dragged to the right by [mathjaxinline]a[/mathjaxinline] units, with zero filling in behind. </p><p>
Both of these operations are important in modeling, and we should work out their Laplace transforms in terms of the Laplace transform of [mathjaxinline]\, f(t) \,[/mathjaxinline] (and the number [mathjaxinline]a\geq 0[/mathjaxinline]). Let's do the delay first: </p><table id="a0000000739" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t-a)\rightsquigarrow \int _0^\infty u(t-a)f(t-a)e^{-st}\, dt[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Since [mathjaxinline]u(t-a)=0[/mathjaxinline] for [mathjaxinline]t<a[/mathjaxinline], we can use [mathjaxinline]t=a[/mathjaxinline] for the lower limit and drop the factor [mathjaxinline]u(t-a)[/mathjaxinline]. </p><table id="a0000000740" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t-a) \rightsquigarrow \int _ a^\infty f(t-a)e^{-st}\, dt \, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This integral calls for the change of variables </p><table id="a0000000741" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\tau =t-a\, ,\quad t=\tau +a\, ,\quad d\tau =dt[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
so the lower limit of [mathjaxinline]\tau[/mathjaxinline] is now 0, and </p><table id="a0000000742" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t-a)\rightsquigarrow \int _0^\infty f(\tau )e^{-s(\tau +a)}\, d\tau \, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
The integral is over [mathjaxinline]\tau[/mathjaxinline], and [mathjaxinline]s[/mathjaxinline] is constant, so we can pull out the factor [mathjaxinline]e^{-as}[/mathjaxinline]: </p><table id="a0000000743" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t-a)\rightsquigarrow e^{-as}\int _0^\infty f(\tau )e^{-s\tau }\, d\tau \, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Do you recognize this integral? It's just the definition of [mathjaxinline]F(s)[/mathjaxinline] (using [mathjaxinline]\tau[/mathjaxinline] instead of [mathjaxinline]t[/mathjaxinline] for the variable of integration). So we have a new rule: </p><p><b class="bfseries"><span style="color:#0000FF"> The [mathjaxinline]t[/mathjaxinline]-shift (or [mathjaxinline]t[/mathjaxinline]-translation) rule:</span></b></p><table id="a0000000744" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000745"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s)\, .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.7)</td></tr></table><p>
Now we turn our attention to finding the Laplace transform of [mathjaxinline]u(t-a)f(t)[/mathjaxinline]. We do this by applying the [mathjaxinline]t[/mathjaxinline]-shift formula to the function [mathjaxinline]g(t)[/mathjaxinline] defined by </p><table id="a0000000746" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]g(t)=f(t+a), \qquad \text { that is, }\qquad g(t-a)=f(t)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
So </p><table id="a0000000747" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)f(t)=u(t-a)g(t-a)\rightsquigarrow e^{-as}\mathcal{L}(g(t);s)=e^{-as}\mathcal{L}(f(t+a);s),.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
This is a second form of the [mathjaxinline]t[/mathjaxinline]-shift rule. The first form expressed [mathjaxinline]\mathcal{L}(u(t-a)f(t-a))[/mathjaxinline] nicely in terms of [mathjaxinline]F(s)[/mathjaxinline]. Unfortunately, there is no such nice general expression for [mathjaxinline]\mathcal{L}(u(t-a)f(t);s)[/mathjaxinline], so we must express it in terms of [mathjaxinline]\mathcal{L}(f(t+a);s)[/mathjaxinline]. </p><p><p><b class="bfseries">Example 5.1 </b> Letting [mathjaxinline]f(t) = 1[/mathjaxinline], and using the [mathjaxinline]t[/mathjaxinline]-shift rule, we find that </p><table id="a0000000748" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)\rightsquigarrow \frac{e^{-as}}{s}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></p><p><p><b class="bfseries">Example 5.2 </b> Suppose [mathjaxinline]\, f(t)=\cos (\omega t)\,[/mathjaxinline] and [mathjaxinline]a>0[/mathjaxinline]. Then </p><table id="a0000000749" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)\cos (\omega (t-a)) \rightsquigarrow e^{-as}\mathcal{L}(\cos (\omega t);s) =\frac{s e^{-as}}{s^2+\omega ^2}\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></p><p><p><b class="bfseries">Example 5.3 </b> On the other hand, to compute [mathjaxinline]\mathcal{L}(u(t-a)\cos (\omega t);s)[/mathjaxinline] we need to use a trig identity. Let [mathjaxinline]\phi =-\omega a[/mathjaxinline], so that </p><table id="a0000000750" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\cos (\omega (t+a))=\cos (\omega t-\phi ) =\cos (\phi )\cos (\omega t)+\sin (\phi )\sin (\omega t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
and hence </p><table id="a0000000751" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]u(t-a)\cos (\omega t) \rightsquigarrow e^{-as}\mathcal{L}(\cos (\omega (t+a));s) =e^{-as}\mathcal{L}(\cos (\phi )\cos (\omega t)+\sin (\phi )\sin (\omega t);s) =e^{-as}\frac{\cos (\phi ) s+\sin (\phi )\omega }{s^2+\omega ^2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table></p><h3 class="hd hd-3 problem-header">Summary:</h3> We have the [mathjaxinline]t[/mathjaxinline]-shift rule expressed in two ways. <table id="a0000000752" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000753"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s)\,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.8)</td></tr><tr id="a0000000754"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a))\, .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.9)</td></tr></table><p><b class="bf">Note 1.</b> This is the first function in the frequency domain that is not a rational function that we have encountered. </p><p><b class="bf">Note 2.</b> [mathjaxinline]e^{-as}[/mathjaxinline] has neither zeros nor poles, for any value of [mathjaxinline]a[/mathjaxinline]. So multiplying by it does not change the pole diagram at all. As far as the pole diagram of the Laplace transform is concerned, the long term behavior of [mathjaxinline]f(t)[/mathjaxinline] is just the same as that of its shift [mathjaxinline]u(t-a)f(t-a)[/mathjaxinline]. </p>
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<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
<h3>Calculations</h3>
</p>
<table id="a0000000755" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000756">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.10)</td>
</tr>
<tr id="a0000000757">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;\mathrm{Re}\, r[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.11)</td>
</tr>
<tr id="a0000000758">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.12)</td>
</tr>
<tr id="a0000000759">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.13)</td>
</tr>
<tr id="a0000000760">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.14)</td>
</tr>
<tr id="a0000000761">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.15)</td>
</tr>
<tr id="a0000000762">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.16)</td>
</tr>
<tr id="a0000000763">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.17)</td>
</tr>
<tr id="a0000000764">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.18)</td>
</tr>
<tr id="a0000000765">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.19)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000000766" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000767">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.20)</td>
</tr>
<tr id="a0000000768">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.21)</td>
</tr>
<tr id="a0000000769">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.22)</td>
</tr>
<tr id="a0000000770">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.23)</td>
</tr>
<tr id="a0000000771">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.24)</td>
</tr>
</table>
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<p>
Find the Laplace transform of </p>
<table id="a0000000772" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]u(t-a)(t-a)^2[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
and </p>
<table id="a0000000773" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]u(t-a)t^2.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
<p style="display:inline">[mathjaxinline]u(t-a)(t-a)^2 \rightsquigarrow[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]u(t-a)t^2 \rightsquigarrow[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">7.6. Examples using the t-shift rule.</h2>
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<h3 class="hd hd-3 problem-header" id="lec7-tab6-problem1-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@lec7-tab6-problem1-problem-progress" tabindex="-1">
Practice problem 1
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@lec7-tab6-problem1-problem-progress"></div>
<div class="problem">
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<p>
<div class="hideshowbox">
<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
<h3>Calculations</h3>
</p>
<table id="a0000000778" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000779">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.25)</td>
</tr>
<tr id="a0000000780">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;\mathrm{Re}\, r[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.26)</td>
</tr>
<tr id="a0000000781">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.27)</td>
</tr>
<tr id="a0000000782">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.28)</td>
</tr>
<tr id="a0000000783">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.29)</td>
</tr>
<tr id="a0000000784">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.30)</td>
</tr>
<tr id="a0000000785">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.31)</td>
</tr>
<tr id="a0000000786">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.32)</td>
</tr>
<tr id="a0000000787">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.33)</td>
</tr>
<tr id="a0000000788">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.34)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000000789" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000790">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.35)</td>
</tr>
<tr id="a0000000791">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.36)</td>
</tr>
<tr id="a0000000792">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.37)</td>
</tr>
<tr id="a0000000793">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.38)</td>
</tr>
<tr id="a0000000794">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.39)</td>
</tr>
</table>
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<a href="javascript: {return false;}">Show</a>
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<p>
Find [mathjaxinline]\mathcal{L}\left(u(t-3)t; s\right).[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]\mathcal{L}\left(u(t-3)t; s\right)=[/mathjaxinline]</p>
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<h3 class="hd hd-3 problem-header" id="lec7-tab6-problem2-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@lec7-tab6-problem2-problem-progress" tabindex="-1">
Practice problem 2
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<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
<h3>Calculations</h3>
</p>
<table id="a0000000796" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000797">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.40)</td>
</tr>
<tr id="a0000000798">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;\mathrm{Re}\, r[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.41)</td>
</tr>
<tr id="a0000000799">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.42)</td>
</tr>
<tr id="a0000000800">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.43)</td>
</tr>
<tr id="a0000000801">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.44)</td>
</tr>
<tr id="a0000000802">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.45)</td>
</tr>
<tr id="a0000000803">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.46)</td>
</tr>
<tr id="a0000000804">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.47)</td>
</tr>
<tr id="a0000000805">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.48)</td>
</tr>
<tr id="a0000000806">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.49)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000000807" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000808">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.50)</td>
</tr>
<tr id="a0000000809">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.51)</td>
</tr>
<tr id="a0000000810">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.52)</td>
</tr>
<tr id="a0000000811">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.53)</td>
</tr>
<tr id="a0000000812">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.54)</td>
</tr>
</table>
</div>
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<a href="javascript: {return false;}">Show</a>
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</p>
<p>
Find [mathjaxinline]\mathcal{L}(f)[/mathjaxinline] for [mathjaxinline]\displaystyle f(t) = \begin{cases} \cos t &amp; 0 &lt; t &lt; 2\pi \\ 0 &amp; t &gt; 2\pi .\end{cases}[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]\mathcal{L}(f)=[/mathjaxinline]</p>
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\(\)
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<tr class="fiptitle">
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
<div class="hideshowcontent">
<p>
<h3>Calculations</h3>
</p>
<table id="a0000000816" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000817">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.55)</td>
</tr>
<tr id="a0000000818">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;\mathrm{Re}\, r[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.56)</td>
</tr>
<tr id="a0000000819">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.57)</td>
</tr>
<tr id="a0000000820">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.58)</td>
</tr>
<tr id="a0000000821">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.59)</td>
</tr>
<tr id="a0000000822">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.60)</td>
</tr>
<tr id="a0000000823">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.61)</td>
</tr>
<tr id="a0000000824">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.62)</td>
</tr>
<tr id="a0000000825">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.63)</td>
</tr>
<tr id="a0000000826">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.64)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000000827" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000828">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.65)</td>
</tr>
<tr id="a0000000829">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.66)</td>
</tr>
<tr id="a0000000830">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.67)</td>
</tr>
<tr id="a0000000831">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.68)</td>
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<tr id="a0000000832">
<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle u(t-a)f(t)[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
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&#160;
</td>
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[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.69)</td>
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<p>
Find [mathjaxinline]\displaystyle \mathcal{L}^{-1}\left( \frac{ e^{-as}}{\frac{s^2}{\omega }+\omega }\right).[/mathjaxinline] </p>
<p>
(Type <b class="bf">omega</b> for [mathjaxinline]\omega[/mathjaxinline]. We provide the multiplication by the appropriate step function.) </p>
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<span class="trailing_text" id="trailing_text_lec7-tab7-problem1_2_1">[mathjaxinline]\cdot u(t-a)[/mathjaxinline]</span>
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<h2 class="hd hd-2 unit-title">7.8. Step response.</h2>
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<p>
It's typical to study a system by studying its response, from rest initial conditions, to a unit step input signal. This is the <b class="bf"> unit step response</b> . In fact this is what we observed in the behavior of the Mascot: the drive shaft moved abruptly to a different position, where it stayed, and the system oscillated in response. </p><p>
In terms of block diagrams, we could write the following. </p><center><img src="/assets/courseware/v1/746c92c4146ad8cea8bfb9e69363b43b/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_stepresponse_block2.svg" width="200px" style="margin: 0px 10px 10px 10px"/></center><p>
We can rewrite this block diagram in the frequency domain. Since [mathjaxinline]u(t) \rightsquigarrow 1/s[/mathjaxinline], if [mathjaxinline]H(s)[/mathjaxinline] is the system function, then the Laplace transform of the step response is [mathjaxinline]H(s)/s[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/39173566ec11febaccd74987d32374ac/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_stepresponse_block1.svg" width="200px" style="margin: 0px 10px 10px 10px"/></center><p><p><b class="bfseries">Example 8.1 </b> Suppose you get a job at time [mathjaxinline]t=0[/mathjaxinline]. Your salary is [mathjaxinline]$S[/mathjaxinline]/month, and you have a total of [mathjaxinline]$E[/mathjaxinline] expenses and taxes each month. Your net income is now [mathjaxinline]$S-E=A[/mathjaxinline]/month. Fortunately your salary is high enough that [mathjaxinline]A[/mathjaxinline] is positive! So you open a bank account and your employer deposits money directly into the account at a rate of [mathjaxinline]A[/mathjaxinline]. </p><p>
The differential equation modeling your bank account with interest rate [mathjaxinline]r[/mathjaxinline] is </p><table id="a0000000836" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\dot x - r x = A\cdot u(t),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]u(t)[/mathjaxinline] is the unit step function. (Your employer is kind enough to continually put money into your account at this rate.) </p><p>
Let [mathjaxinline]t=0[/mathjaxinline] be the time you start the job. At [mathjaxinline]t=0[/mathjaxinline] the bank account is empty (rest initial conditions). </p><p>
Find the step response [mathjaxinline]x[/mathjaxinline] to the differential equation above with step input [mathjaxinline]A\cdot u(t)[/mathjaxinline] using rest initial conditions. </p></p><p><b class="bfseries"><span style="color:#FF7800">Solution:</span></b> Taking the Laplace transform of both sides, we get </p><table id="a0000000837" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000838"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle (s-r)X[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A/s[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.71)</td></tr><tr id="a0000000839"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle X[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{A}{s(s-r)}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.72)</td></tr></table><p>
Using coverup, we write this in terms of partial fractions: </p><table id="a0000000840" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000841"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{A}{s(s-r)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{-A/r}{s} + \frac{A/r}{s-r}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(5.73)</td></tr></table><p>
Taking the inverse Laplace transform, and using the [mathjaxinline]s[/mathjaxinline]-shift rule or inverse table lookup, we get </p><table id="a0000000842" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]x(t) =Au(t)\left( -\frac{1}{r} + \frac{1}{r}e^{rt} \right).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><img src="/assets/courseware/v1/3c885704c23fe71e755c56cf37bf3baa/asset-v1:OCW+18.031+2019_Spring+type@asset+block/images_c5_stepresponse_bank.svg" width="450px" style="margin: 0px 10px 10px 10px"/><br/><font size="2">Step response for [mathjaxinline]r=.01[/mathjaxinline].</font><br/></center><div><br/></div><p>
More complicated examples can be built out of step response and delay. </p>
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