 #1
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 Homework Statement:
 I need help with finding the approximate solution of a second order differential equation.
 Relevant Equations:

I am given the differential equation as in
$$
\lambda \frac{d^2u}{dx^2} + q = 0
$$
which is defined over $$x \in [L/2,L/2]$$ and where $$q = a+bx$$. I am supposed to find an approximative solution using a spectral method, i.e. using cosine and sine terms that fulfills the essential boundary conditions given by
$$
u \bigg ( \frac{L}{2} \bigg ) = 0, u \bigg (\frac{L}{2} \bigg ) = 0
$$
The appoximative solution should be represented by a Fourier series and by noticing that the terms in the integral are orthogonal, the matrix of the system of equations should be diagonal.
We choose an approximative solution given by
$$
u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx
$$
Comparing this approximative solution with the differential equation yields that
$$
\frac{a_0}{2} = a
$$
and the boundary conditions yields the equation system
$$
a + \sum_{n=1}^N a_n \cos \bigg ( \frac{nL}{2} \bigg ) + b_n \sin \bigg ( \frac{nL}{2} \bigg ) = 0 \\
a + \sum_{n=1}^N a_n \cos \bigg ( \frac{nL}{2} \bigg ) + b_n \sin \bigg ( \frac{nL}{2} \bigg ) = 0
$$
So at least I got a system of two equations but I do not know where to go from here. How should I use the orthogonality to set up a matrix from these two equations?
$$
u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx
$$
Comparing this approximative solution with the differential equation yields that
$$
\frac{a_0}{2} = a
$$
and the boundary conditions yields the equation system
$$
a + \sum_{n=1}^N a_n \cos \bigg ( \frac{nL}{2} \bigg ) + b_n \sin \bigg ( \frac{nL}{2} \bigg ) = 0 \\
a + \sum_{n=1}^N a_n \cos \bigg ( \frac{nL}{2} \bigg ) + b_n \sin \bigg ( \frac{nL}{2} \bigg ) = 0
$$
So at least I got a system of two equations but I do not know where to go from here. How should I use the orthogonality to set up a matrix from these two equations?