# Second order differential equation

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?

BvU
Homework Helper
Comparing this approximative solution with the differential equation yields that
##{a_0\over 2}=a##
How ?

and the boundary conditions yields the equation system
I miss the ## +bx## from
$$\lambda \frac{d^2u}{dx^2} + a + bx = 0$$

(and I find the use of ##a_n## and ##b_n## in your notation confusing, since ##a## and ##b## are allready used).

pasmith
Homework Helper
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
$$\frac{a_0}{2} = a$$

No. It's the second derivative of $u$ that is equal to $\frac{-q(x)}{\lambda}$, not $u$ itself. When you differentiate the constant term you get zero, and in any case a non-zero constant does not satisfy the boundary condition.

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?

You have $u(x) \approx \sum_n u_n\phi_n(x)$ where $$\phi_n(-L/2) = \phi_n(L/2) = 0.$$ Therefore $$\lambda \sum_n u_n \phi_n''(x) \approx -q(x)$$ and we determine the $u_n$ by requiring that the inner product with $\phi_m$ should be exact, ie. $$\lambda \sum_n u_n \int_{-L/2}^{L/2} \phi_n''(x) \phi_m(x)\,dx = -\int_{-L/2}^{L/2} q(x)\phi_m(x)\,dx.$$ The matrix you are looking for is the matrix of coefficients $$M_{mn} = \int_{-L/2}^{L/2} \phi_n''(x) \phi_m(x)\,dx.$$

Now the choice of a fourier basis means that $$\phi_n(x) = A_n\cos(k_n x) + B_n\sin(k_n x)$$ and $k_n$ is such that the boundary conditions are satisfied by something other than $A_n = B_n = 0$.

Last edited:
etotheipi