# General solution to inhomogeneous second order equation

## Homework Statement

I need to find the solution to x'' + cx' = f(t), for a general f.

## The Attempt at a Solution

Obviously first I solve the homogeneous part to give me A + B*exp(-ct). I also know that the particular solution is written as (1/c)int((1-exp(c(s-t))f(s))ds where the integral is between 0 and t. However I am not sure why this is so, any help would be much appreciated.

gabbagabbahey
Homework Helper
Gold Member
Have you learned about Fourier Transforms yet? If so, just transform both sides of the DE, solve the resulting algebraic equation for $\tilde(x)(t)$ and then take the inverse Fourier Transform.

HallsofIvy
Homework Helper
Oh, dear! Using "Fourer Series" for this is like using a shotgun to kill a fly!

Let v= x' and your differential equation becomes v'+ cv= f(t). That's a linear equation with "integrating factor" $e^{ct}$. That is,
$$\frac{d(e^{ct}v)}{dx}= e^{ct}v'+ ce^{ct}v= e^{ct}f(t)$$

Integrating both sides,
$$e^{ct}v= \int_{t_0}^t e^{cs}f(x)ds+ C$$

From that,
$$v= x'= Ce^{-ct}+ e^{-ct}\int_{t_0}^t e^{cs}f(s)ds$$

Now, integrate again:
$$x(t)= C_1 e^{-ct}+ \int_{t_0}^t\left(e^{-cu}\int_{t_0}^u e^{cs}f(s)ds\right)du+ C_2$$