I'm trying to follow a proof for the solution of the diffusion equation in 0 < x < l with inhomogeneous boundary conditions.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]u_n(0) = 0[/tex]

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

[tex] 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 [/tex]

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# What kind of ODE is this?

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