How to Prove Series Solutions for Differential Equations?

[popo902]

Homework Statement

I was wondering, how you would prove that the solutions work for an equation?
i know for a normal Diff eq, you just plug your solutions back into the equation
but how would i go about showing that a series solution IS a solution to a problem?

For instance, if they gave you two linearly independent solutions and told you to show that they were solutions of an equation.

Homework Equations

The Attempt at a Solution

 

[hunt_mat]

You do the same thing as before, you plug your series solution back into your equation and what will happen is that the terms start cancelling out. Try this for yourself, take a simple ODE say:

$$\frac{d^{2}y}{dx^{2}}+y=0$$

Take the series representation for either sin or cos and see what happens.

 

[popo902]

oh i see now
just some simplification of the series combination
and it does cancel
Thank you

 

[hunt_mat]

That's what I'm here for...