General solution to inhomogeneous second order equation

[NT123]

Homework Statement

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

Homework Equations

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]

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]

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