# Fourth-order homogeneous differential equation with polynomial coefficients

Suppose I have a differential equation

\begin{align} \sum_{i=0}^4 f_i (x) y^{(i)}(x) = 0, \end{align} where $$y^{(i)}(x)$$ represents the $$i$$th derivative of $$y(x)$$ and the coefficient functions $$f_i(x)$$ are all polynomials of the variable $$x$$. Suppose also there are some boundary conditions, although for the purposes of this question they may not be important.

For special cases of the $$f_i$$'s, there are exact analytic solutions to this differential equation. However, in general, there are no exact solutions that can be found.

Are there any techniques to arriving at an approximate solution $$y(x)$$ with arbitrary polynomial coefficient functions $$f_i(x)$$?

• Have you considered the power series method ? – Yves Daoust Jan 23 at 23:09