What are the numerical or analytical solution methods (preferably analytical) for solving following differential equation to find $A_n(t)$? and how to solve it?
$$B.A_n''(t)+(\frac{n\pi}{l})^4A_n(t)$$ $$+C\sum_{i=1}^N\bigg[\bigg(\frac{2}{l}\sin(\frac{i\pi vt}{l})\sin(\frac{n\pi vt}{l})\bigg)\bigg(1+\frac{i\pi}{l}A_i'(t)+v'\frac{i\pi}{l}A_i(t)+(\frac{i\pi}{l})^2A_i(t)+G(t)\bigg)\bigg]=F_n(t)$$ where $B,C,l$ are constants, $v(t)$ is velocity, $F_n(t)$ and $G(t)$ are functions of $t$.
Is there anybody who has idea to solve?