# Characterizing second order ODE

1. Jan 13, 2006

### saltydog

Regarding:

$$(a+bx+cx^2)y^{''}+(f+gx+hx^2)y^{'}+(j+kx+mx^2)y=0$$

Does anyone here know if it's been "completely" characterized in terms of the geometry of the three parabolas which make up it's coefficients?

For example, if I'm given plots of the parabolas, can any information at all be extracted from them in order to determine at the very least the general appearance of the solution of the corresponding DE without having to directly solve it?