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<h2 class="hd hd-2 unit-title">11.1. Activity: Radioactive material.</h2>
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<h3 class="hd hd-2">Convolution in real life</h3>
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<p><b class="bfseries"><span style="color:#FF7800">Note on video:</span></b> In the video above, we discuss an example using convolution, which doesn't appear to have anything to do with differential equations. However, embedded in this example is a differential equation. In the following activity, we bring out the differential equation. The sequence of questions in this activity review the Laplace transform methods you have learned in the previous lectures, and compare this to the convolution method we've just introduced. Along the way, we will see why the unit impulse response is also called the weight function of the system. </p>
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Suppose a vessel is loaded with radioactive material at a rate given by [mathjaxinline]f(t)=e^{-at}[/mathjaxinline], starting with an empty vessel at [mathjaxinline]t=0[/mathjaxinline]. By coincidence, the decay rate of the material is given by the same constant [mathjaxinline]a[/mathjaxinline] for [mathjaxinline]a&gt;0[/mathjaxinline]. </p>
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Set up the differential equation modeling the amount of radioactive material [mathjaxinline]x[/mathjaxinline] in this system. Use Laplace transform methods to find an expression for [mathjaxinline]X[/mathjaxinline] in terms of [mathjaxinline]s[/mathjaxinline] and [mathjaxinline]a[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]X(s)=[/mathjaxinline]</p>
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<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
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Activity (b).
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Solve for [mathjaxinline]x[/mathjaxinline] by finding the inverse Laplace transform of [mathjaxinline]X[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_rec11-tab1-problem2_2_1">[mathjaxinline]\cdot u(t)[/mathjaxinline]</span>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
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<td class="formulainput">^ (raise to a power)</td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<tr class="formulainput">
<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>
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<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>
</tr>
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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>
<tr class="formulainput">
<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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<h3 class="hd hd-3 problem-header" id="rec11-tab1-problem3-problem-title" aria-describedby="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec11-tab1-problem3-problem-progress" tabindex="-1">
Activity (c).
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What is the transfer function [mathjaxinline]H(s)[/mathjaxinline] for this system with input [mathjaxinline]f(t)=e^{-at}[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]H(s)=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<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>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
<tr class="formulainput">
<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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Activity (d).
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<p>
What is the weight function [mathjaxinline]w(t)[/mathjaxinline]? (Recall, weight function is another name for the unit impulse response.) </p>
<p>
<p style="display:inline">[mathjaxinline]w(t)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_rec11-tab1-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<span class="trailing_text" id="trailing_text_rec11-tab1-problem4_2_1">[mathjaxinline]\cdot u(t)[/mathjaxinline]</span>
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\(\)
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<div class="formulainput">
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<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<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>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
<tr class="formulainput">
<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>
</tbody>
</table>
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Activity (e).
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Recall that [mathjaxinline]\, f(t) = e^{-at}[/mathjaxinline]. Compute the convolution [mathjaxinline]f(t)*w(t)[/mathjaxinline], where [mathjaxinline]\, w(t)\,[/mathjaxinline] is the weight function you found in part <b class="bf">(d)</b>. (Does this agree with your answer to <b class="bf">(b)</b>?) </p>
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<p style="display:inline">[mathjaxinline]f(t)*w(t)=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_rec11-tab1-problem5_2_1">[mathjaxinline]\cdot u(t)[/mathjaxinline]</span>
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Activity (f).
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<p>
In the mathlet above set [mathjaxinline]f(t)=e^{-at}[/mathjaxinline] and [mathjaxinline]g(t)=e^{-at}[/mathjaxinline], and estimate the time [mathjaxinline]t[/mathjaxinline] at which the convolution achieves a maximum. Given that [mathjaxinline]a=\ln (2)[/mathjaxinline], compare this with a calculation using your computation of the convolution. </p>
<p>
(Try to get within 1/8 of the exact answer. <b class="bfseries"><span style="color:#FF7800">Pro tip:</span></b> You can move the [mathjaxinline]u[/mathjaxinline] slider to get the value of [mathjaxinline]t[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]t_{\textrm{max}}=[/mathjaxinline]</p>
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