For a 2nd order linear inhomogeneous ODE

\(\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)\)

where \(f(x)\) is a periodic function that given the period \(L_{0}\).
To find the ODE's particular integral,
1. Calculate the Fourier series of the function \(f(x)\)
2. Try ansatz and using the period \(L_{0}\): \(\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)\).
3. Yield the derivatives of \(y(x)\) and plug them into the original ODE.
4. Comparing coefficients in front of 1, cosine, sine, respectively, write out the equation system about the ODE.
5. Solve the equation system and write out the particular integral.