Differential Equations with Dicontinuous Forcing Functions

[louis11]

Homework Statement

Find the solution of the given initial value problem.

Homework Equations

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

$$f(t) = \left\{ \begin{array}{c l} 1, & 0 \le t \le \pi/2 \\ 0, & \pi/2 \le t < \infty \end{array} \right.$$

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

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:

1. Find the step function
2. Set the differential equation equal to the step function
3. Find the Laplace transform of both sides
4. 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?

 

[mighty2000]

Hello,

Thank you for your post. It seems like you are on the right track with your approach. Here are some suggestions:

1. Instead of using the step function, you can break the problem into two separate cases. One for t between 0 and pi/2, and one for t greater than pi/2. This will make the calculations a bit simpler.

2. In the first case (0 to pi/2), the differential equation becomes y" + y = 1. This is a simple equation that can be solved using standard methods.

3. In the second case (pi/2 to infinity), the differential equation becomes y" + y = 0. This is a homogeneous equation that can be solved using standard methods.

4. Once you have the solutions for both cases, you can use the initial conditions (y(0) = 0, y'(0) = 1) to determine the constants of integration.

5. As for the shifting of the t variable, it is used to take into account the change in the function at a certain point, such as a boundary in a piecewise function. In this problem, you do not need to shift the t variable because the initial conditions are given at t = 0.

I hope this helps. Let me know if you have any further questions.
Scientist