Represent the following function using unit step functions and find its Laplace transform. f(x)={2x2,0&lt;x&lt;2x+3,x&gt;2

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Represent the following function using unit step functions and find its Laplace transform. f(x)={2x2,0&amp;lt;x&amp;lt;2x+3,x&amp;gt;2

dessinemoie · Accepted Answer

Step 1: Introduction1. Piecewise functions are those whose function expression varies depending on where in their domain they occur.2. Unit step functions are functions with values of zero for negative input and one for positive input.Step 2: ComputationPiecewise function&amp;nbsp;f(x)={f1(x),0≤x&amp;lt;x1f2(x),x1≤x&amp;nbsp;&amp;nbsp;, can be written in unit step function asf(x)=f1(x)+u(x−x1)(f2(x)−f1(x))So, the given function&amp;nbsp;f(x)={2x2,0&amp;lt;x&amp;lt;2x+3,x&amp;gt;2&amp;nbsp;&amp;nbsp;can be written in unit step function asf(x)=2x2+u(x−2)(x+3−2x2)Step 3:ComputationNow, to find Laplace transform of given function f(x), solve the equationL(f)=L(2x2)+L[u(x−2)(x+3−2x2)]. Use the property&amp;nbsp;L(u(t−a)f(t))=e−st⋅L(g(t+a))&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;L(tn)=n!sn+1L(f)=L(2x2)+L[u(x−2)(x+3−2x2)]=2L(x2)+L(u(x−2)(x+3−2x2))=2(2!s2+1)+e−2sL(x+2)+3−2(x+2)2=4s3+e−2sL(x+5−2x2−8−8x)=4s3+e−2sL(−2x2−3−7x)=4s3+e−2s(−2(2!s2+1)−3s−7s2)&amp;nbsp;

Represent the following function using unit step functions and find its Laplace transform.f(x)={(2x^2, 0<x<2),(x+3, x > 2):}

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