Homework Help: 2nd Order ODE Initial Value Proof Problem

1. Jul 7, 2013

DuncP89

1. Check that

y(t)=1/λ ∫_0-t_〖f(s) *sin(λ(t-s) )ds〗
is the solution of the following initial value problem

y''(t)+λ^2y(t)=f(t), λ>0, y(0)=0,y'(0)=0

2. Relevant equations

3. I tried to do integration by parts on y(t), but just doesn't work. I'm not sure how to prove it using those two initial conditions

2. Jul 7, 2013

SteamKing

Staff Emeritus
Try using Laplace transforms.

3. Jul 7, 2013

DuncP89

I have never learned laplace transformation, I'm sure my prof is not looking for anything like that

4. Jul 7, 2013

epenguin

You are not asked to integrate or invent Laplace transforms, only to differentiate something, with the advantage of being told the answer.

Last edited: Jul 7, 2013
5. Jul 8, 2013

DuncP89

Forgive me, I don't quite understand it. How is it only about differentiate? There is an integral on the answer y(t), you can get rid of it on y''(t), but I don't know how to get rid of it on λ^2*y(t)

6. Jul 8, 2013

SteamKing

Staff Emeritus
Obviously, you can't. All you can do is multiply λ^2 by y(t).

7. Jul 8, 2013

vanhees71

I haven't tried it myself, but if I understand the question correctly, you should not try to do the integral but just take the derivatives wrt. $t$ and prove that it fulfills the differential equation and the initial conditions.

On the other hand, that's pretty clear since, what's written there is nothing than the convolution of the Green's function of the undamped oscillator-differential operator with the inhomogenity on the right-hand side, and thus the claim in the problem is correct.

8. Jul 8, 2013