2nd Order ODE Initial Value Proof Problem

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

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SteamKing
Staff Emeritus
Homework Helper
Try using Laplace transforms.

Try using Laplace transforms.
I have never learned laplace transformation, I'm sure my prof is not looking for anything like that

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

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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
Staff Emeritus
Homework Helper
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.