Understanding and Solving ODEs with Inhomogeneous Boundary Conditions

[StewartHolmes]

I'm trying to follow a proof for the solution of the diffusion equation in 0 < x < l with inhomogeneous boundary conditions.

\frac{d u_n(t)}{dt} = k( -\lambda_n u_n(t) - \frac{2n\pi}{l}[ (-1)^n j(t) - h(t) ] )
u_n(0) = 0

Now I just plain don't understand what kind of an ODE I have here. If the term in j(t) and h(t) wasn't there, it'd be a simple ODE, but I'm confused as to what can be done now. I know ODEs of the form

y' + p(x)y + q(x) = 0

But I have something like, y' + p(x)y + q(t) where I have a term in the dependent variable.

The book I have gives the solution as
u_n(t) = Ce^{-\lambda_n kt} - \frac{2n\pi k}{l}\int\limits_0^t e^{-\lambda_n k(t-s)} \left( (-1)^n j(s) - h(s) \right) \, ds

 

[jackmell]

Try and learn to encapsulate everything. You have:

<br /> \frac{d u_n(t)}{dt} = k( -\lambda_n u_n(t) - \frac{2n\pi}{l}[ (-1)^n j(t) - h(t) ] )<br />

Now, isn't the term:

-\frac{2n\pi}{l}k[(-1)^n j(t)-h(t)]

just some function of t? Say it's v(t). So you have essentially the equation:

\frac{dy}{dt}+k\lambda y=v(t)

And you know how to integrate that right by finding the integration factor. Change it to u_n if you want, but it's the same equation essentially.