2nd Order ODE Initial Value Proof Problem

[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

Homework 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

 

[SteamKing]

Try using Laplace transforms.

 

[DuncP89]

Try using Laplace transforms.

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

 

[epenguin]

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

 

[DuncP89]

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

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)

 

[SteamKing]

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

 

[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.

 

[ehild]

Find the second derivative of the given y(t) function, and then plug in y"and y(t) into the differential equation to see if y(t) is a solution or not.
How to differentiate a definite integral? See http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

ehild