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<h2 class="hd hd-2 unit-title">PS3.1. Step functions</h2>
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Find the Laplace transform of:<br/></p>
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(a) The function given by [mathjaxinline]\sin (\omega t)[/mathjaxinline] for [mathjaxinline]0&lt;t&lt;2\pi[/mathjaxinline] and 1 otherwise. </p>
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(b) The function given by [mathjaxinline]t^2[/mathjaxinline] for [mathjaxinline]t&gt;2[/mathjaxinline] and 0 otherwise. </p>
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<h2 class="hd hd-2 unit-title">PS3.2. Step response</h2>
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(a) What is the Laplace transform of the Heaviside step function [mathjaxinline]u(t)[/mathjaxinline]? </p>
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(b) What is the Laplace transform of the derivative of the Heaviside function? </p>
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(c) What is the Laplace transform of the unit step response of an LTI system with system function [mathjaxinline]H(s)[/mathjaxinline]? </p>
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(d) Consider the system [mathjaxinline]\dot{x}+2x = f(t)[/mathjaxinline], with input [mathjaxinline]f[/mathjaxinline] and response [mathjaxinline]x[/mathjaxinline]. Find the unit step response. (Don't forget to multiply by an appropriate step function.) </p>
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(e) What is the long-term behaviour of this response? </p>
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<h2 class="hd hd-2 unit-title">PS3.3. Step and impulse response</h2>
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Give the step and impulse responses for the system [mathjaxinline]m \ddot{x} + b \dot{x} + k x[/mathjaxinline] with [mathjaxinline]m=2[/mathjaxinline], [mathjaxinline]b=8[/mathjaxinline] and [mathjaxinline]k=10[/mathjaxinline]. Sketch [mathjaxinline]x(t)[/mathjaxinline] and [mathjaxinline]\dot{x}(t)[/mathjaxinline] in each case. </p>
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(Type <b class="bf">u(t-1) </b> for the unit step function shifted by 1. Type <b class="bf">delta(t)</b> for the delta function.) </p>
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<p style="display:inline">Step response: &#8195;</p>
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<p style="display:inline">Impulse response: &#8195;</p>
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<h2 class="hd hd-2 unit-title">PS3.4. Fish example revisited</h2>
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4
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<p>
<span style="color:#FF0000">The following problems do not have checkers.</span>
</p>
<p>
In the fish population example from the online reading, the program is refunded each year and they have enough money to restock at a constant rate of [mathjaxinline]r[/mathjaxinline] for the first half of each year. Find [mathjaxinline]x(t)[/mathjaxinline] in this case. </p>
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5
</h3>
<div class="problem-progress" id="block-v1:OCW+18.031+2019_Spring+type@problem+block@ps3-tab4-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
In the fish population example from the online reading, now suppose that at the beginning of each year [mathjaxinline]r/2[/mathjaxinline] units of fish are dumped all at once into the lake. Find [mathjaxinline]x(t)[/mathjaxinline] in this case. </p>
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