(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the solution of the given initial value problem.

2. Relevant equations

y" + y = f(t); y(0) = 0, y'(0) = 1;

[tex]

f(t) = \left\{

\begin{array}{c l}

1, & 0 \le t \le \pi/2 \\

0, & \pi/2 \le t < \infty

\end{array}

\right.

[/tex]

Edit: That should be an f(t). Not an f(x) in the piecewise function. For whatever reason it defaults to the "x".

3. The attempt at a solution

My book provides one example regarding this type of problem. An example that I have been trying to follow for about the last hour to no avail. I believe the steps required to solve this problem are as follows:

- Find the step function
- Set the differential equation equal to the step function
- Find the Laplace transform of both sides
- Separate like terms. Find the inverse Laplace to solve for y(t)

I don't know if these steps are correct but it's what I took from the example provided. Anyway, below is what I've attempted:

L{y"+y} -> [s^2Y(s)-sy(0)-y'(0)] + Y(s) = f(t)

f(t) = 1 - u_pi/2(t)

Y(s) = [(1 - u_pi/2(t)) + 1] / (s^2 + 1)

At this point when I try and take the Laplace the answer I get isn't correct. It seems like it's on the right track, but I'm missing some element along the way that throws it off slightly.

Yesterday when I was working with step functions I had some difficulty with the "shifting". For some problems, to get the correct answer I had to "shift" the t by some factor, usually a boundary in the piecewise function. In this problem, do I need to shift? What's the purpose of the shift, and how do you know when you should do it?

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# Homework Help: Differential Equations with Dicontinuous Forcing Functions

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