# Solution of differential equation with polynomial coefficients

I have come across a rather general form of differential equation and I was wondering if anyone knew what they would be called or where I can find some literature on them.

Imagine we have some function $f$ and we have a list of polynomials $p_0,p_1,...,p_n \in \mathbb{C}[x]$ and $f$ satisfies the differential equation:

$$\sum_{j=0}^n p_j(x) \frac{d^j f}{dx^j} = 0$$

I would like to know results on bounding $f$ and methods of solving for $f$ on the positive real line. Or even just algorithms of solving for $f$. I would also like to know what we would call these differential equations. I thought "homogeneous linear differential equations with polynomial coefficients" might be close but I was wondering if perhaps there was a more exact name. Or if anyone knows of literature that might cover these differential equations, that would be very helpful.

Any help would be greatly appreciated! Thanks!

• It is nth order homogeneous linear equation. There is no general method for solving DEs with variable coefficients when $n\geq 1$. You may use Power series solutions. – daulomb Oct 29 '13 at 21:05