# 2nd order ODE - Show solution by substitution

## Homework Statement

Show that

y(t) = (1/w) ∫[0,t] f(s)*sin(w(t-s)) ds

is a particular solution to

y'' +w2 y = f(t)

where w is a constant.

## The Attempt at a Solution

After wasting several pages of paper I have made virtually no progress. Obviously, substitution suggests you plug in y(t), differentiate it twice for the first term, and somehow arrive at f(t) = f(t). However, without more information about f(s), it seems impossible. Integration by parts on y(t) will result in another integral which in turn must be integrated by parts...and so on infinitely many times. Therefore integration by parts of the solution y(t) is not an option.

I have considered that this could be an application of the fundamental theorem of calculus, but with an additional sine term that also depends on t in the integrand, it seems not not apply (at least not in a way I am familiar with).

This is a problem from Richard Haberman's text: Applied PDE with Fourier Series and Boundary Value Problems.

Any ideas/advice would be much appreciated.
Thank you

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LCKurtz
Homework Helper
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## Homework Statement

Show that

y(t) = (1/w) ∫[0,t] f(s)*sin(w(t-s)) ds

is a particular solution to

y'' +w2 y = f(t)

where w is a constant.

## The Attempt at a Solution

After wasting several pages of paper I have made virtually no progress. Obviously, substitution suggests you plug in y(t), differentiate it twice for the first term, and somehow arrive at f(t) = f(t). However, without more information about f(s), it seems impossible. Integration by parts on y(t) will result in another integral which in turn must be integrated by parts...and so on infinitely many times. Therefore integration by parts of the solution y(t) is not an option.

I have considered that this could be an application of the fundamental theorem of calculus, but with an additional sine term that also depends on t in the integrand, it seems not not apply (at least not in a way I am familiar with).

This is a problem from Richard Haberman's text: Applied PDE with Fourier Series and Boundary Value Problems.

Any ideas/advice would be much appreciated.
Thank you
You don't need to integrate anything. You need to differentiate using Leibnitz's rule. There are several forms. You will find this form useful:$$\frac d {dt} \int_0^t g(s,t)\, ds = \int_0^t \frac {\partial g(s,t)}{\partial t}\, ds + g(t,t)$$

So as I suspected, it's some rule that I was unfamiliar with. Thanks for the help!