Help solving ODE's using eigenfunction expansion, in general

[ericm1234]

My ODE textbook does not help me much here; and a large number of master's exam practice tests (with worked out solutions) also isn't helping me. I need someone to recommend a (preferably ONLINE) source that clearly states how to solves ODES using eigenfunction expansion.
For example, y''+y=kcosx with y'(0)=0, y'(pie)=1.
This is just an example of the type I want to learn how to solve.
It seems that when the boundary conditions are homogeneous, this problem is easily solved by plugging in an infinite series: y=Sum(a_n*cos(nx)) into the left side, and then plugging in a different series: Sum(b_n*cosnx)=kcosx..then equating coefficients and getting the b_n's. BUT when the BC are non-homogeneous, there appears to be extra work.

Help.

 

[MrHigh]

Uhm... i am not sure you have a proper textbook.
Also, I am not sure if you are familiar with Fourier tranformation. It is an interesting way to deal a wide class of equations.
Basically, it is almost the same thing you did but applying the transformation at both sides . You can easily look in the web. for exemple your example will became after transforming, something like..

$(1-w^2)\tilde{y}(w) = kC (\delta(w-1) + \delta(w+1))$

You see that the solution is straightforward as there is no derivatives. You should now inverse transform the solution if you want. I used some properties of Fourier transform to deal with the derivatives but again i think you should find it easily.

 

[ericm1234]

Ok let me rephrase: can someone give me and work out a complete example with non-homogeneous boundary conditions, using eigenfunction expansion? let's say, y''+y=x+ cosx,
y(0)=0, y(pie)=A..just to come up with one off the top of my head.