**The Laplace Transform**

Part 16: Nonhomogeneous Differential Equations

Another advantage of using Laplace transforms on differential equations is when the equation is nonhomogeneous. For example, consider the problem

, ,

where H(t) is the Heaviside step function, and *f*(*t*) is the square wave . Note that due to our initial conditions, and the presence of the Heaviside step function on the right-hand side, *y*(*t*)=0 for *t*<0.

Taking the Laplace transform of our differential equation, letting ,

Recall from here that we found, using the periodic function rule, that the square wave has Laplace transform , so using the scaling rule, we see that the Laplace transform of our function *f*(*t*) is .

So, to continue with our solution,

,

where we have used the frequency shift formula and the transform for the sine. Now, recalling the convolution rule, we see then that

.

For those who wish to know what this convolution looks like, we continue by breaking up the region of integration piecewise via the square wave:

Below is a graph of this function for over two periods of *f*(*t*),

And here is a close-up for small *t*, showing more clearly that *y*(0)=*y*‘(0)=0.

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Tags: Differential Equation, Laplace Transform, Math, Monday Math, Periodic Function

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