When and How to Solve ODEs: Clarity for Confused Students

[ABearon]

TL;DR

when to use e^mx and x^m

I know how to solve ODEs using both methods. The problem I'm having is knowing when to use one and not the other. If someone could help clarify this for me. I can't find the correct section in my textbook.

 

[BvU]

Hello ABearon, !

Bit hard to answer without examples. Perhaps you can provide an example for yourself by going the other way: try to compose an exercise where $x^m$ is a useful trial function and another example where $e^{mx}$ is a good choice. Not too hard: simply differentiate and see what kind of DE you can make form the result !

 

[ABearon]

Hello ABearon, !

Bit hard to answer without examples. Perhaps you can provide an example for yourself by going the other way: try to compose an exercise where $x^m$ is a useful trial function and another example where $e^{mx}$ is a good choice. Not too hard: simply differentiate and see what kind of DE you can make form the result !

I think i figured it out. We're supposed to use y=e^mx when the ode has constant coefficients (a, b, c) and y=x^m for Cauchy-Euler equations, which are ODEs but the terms have have a-sub-n(x^n)(d^n y/dx^n)

 

[HallsofIvy]

The derivatives of $e^{ax}$ are all of the form $e^{ax}$ again. That is why they are solutions to linear differential equations with constant coefficients. The derivatives of $x^a$ are of the form $x^b$ with b< a. That is why the are solutions to linear differential equations with powers of x as coefficients.