ODE w/ Discontinuous forcing function

[riskybeats]

Homework Statement

My computer is having major issues with latex right now, so sorry for not putting it in latex format.

y'' + y' + (5/4)y = g(t)

y(0) = 0, y'(0) = 0, g(t) = piecewise function {sint, o <= t < pi; 0, t>= pi

Homework Equations

Laplace transform

The Attempt at a Solution

I can get it into the form y(s² + s + (5/4)) = (1+e^-pi*s)/(s² + 1)

When you complete the square of the terms in front of y to the left, it's:
((s+1/2)² + 1), which when divided into the other side is suspiciously like F(s + 1/2) where F(s) = 1/(s² + 1), which would match the other sides nicely as well.

Unfortunately, I don't know how to do it in this form (inverting F(s-2) * F(s)(1+e^-pi*s), or if I'm to do partial fractions with the right hand side.

Any suggestions? Thanks!

 

[mighty2000]

Hi there! Thank you for sharing your problem with us. Based on your attempt at a solution, it seems like you are on the right track. I would suggest using the Laplace transform to solve this differential equation. The Laplace transform is a mathematical tool that can be used to solve linear differential equations with constant coefficients, such as the one in your problem.

To use the Laplace transform, you will need to take the Laplace transform of both sides of the equation. This will give you an algebraic equation in terms of the transformed function, Y(s). Then, you can solve for Y(s) and use the inverse Laplace transform to find the solution, y(t).

In your problem, the Laplace transform of the left side of the equation would be (s^2 + s + 5/4)Y(s), as you have correctly identified. For the right side, you will need to use the Laplace transform of the piecewise function. You can do this by breaking the integral into two parts, one for the interval [0, pi) and one for the interval [pi, ∞). Then, you can use the Laplace transform of each part separately and add them together.

Once you have solved for Y(s), you can use the inverse Laplace transform to find the solution, y(t). This may involve using partial fractions, as you have mentioned. I would recommend checking your algebra carefully and using a table of Laplace transforms to help you with the calculations.

I hope this helps! Let me know if you have any further questions. Good luck with your problem!