I encountered the question for the particular solution of, $$ k \frac{d^4y}{dx^4} = m x $$ where m and k are real numbers. I would solve this question with basic methods for ODEs but question requires $mx$ to be expressed as a Fourier half sine series (I can do it) and asks to find particular solution of the ODE by using that expression.
I wrote the Fourier half sine expression but how can I find particular solution for that? Variation of parameters method looks too long to be the optimal method for this equation since its 4th order and as far as I know undetermined coefficient method cannot be used since Fourier series expression is not suitable for it.