# Solving second order homogenous ODES

Hi there,

I am trying to solve a second order homogenous ODE. I know how to do this when the coefficients of the complimentary polynomial are numbers, but what do I do when they are expressions of x?

Thank you!
(edited 3 weeks ago)
Original post by cata03
Hi there,

I am trying to solve a second order homogenous ODE. I know how to do this when the coefficients of the complimentary polynomial are numbers, but what do I do when they are expressions of x?

e.g. (e^ax)(d2y/dx2) + a(e^ax)(dy/dx) = 0

Thank you!

Its an equation, can you not first divide through by ... (assuming Im reading it correctly)
For your given example, why not divide through by e^ax?
Original post by DFranklin
For your given example, why not divide through by e^ax?

Hahaha that's very true, bad example. In general though, is there a specific method to use these situations or is it purely a case by case basis?
Original post by cata03
Hahaha that's very true, bad example. In general though, is there a specific method to use these situations or is it purely a case by case basis?

It's definitely a different league of difficulty than the constant coefficient case.

The (fairly general) method I covered at university was https://en.m.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory

but unless you're specifically doing something like that, any example you get given will typically have a simple trick (usually a substitution) that reduces it to the constant coefficient case.