How to Prove Series Solutions for Differential Equations?

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Homework Help Overview

The discussion centers around proving that series solutions for differential equations are valid solutions. The original poster seeks clarification on how to demonstrate that given series solutions satisfy a specific differential equation.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the method of substituting series solutions back into the differential equation to verify their validity. There is mention of simplification and cancellation of terms as part of this process.

Discussion Status

The conversation is ongoing, with some participants providing insights into the verification process. There is an acknowledgment of the approach involving substitution and simplification, but no consensus or final resolution has been reached.

Contextual Notes

The original poster references the need to show that two linearly independent solutions are indeed solutions to a differential equation, indicating a specific context for the discussion.

popo902
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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.

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The Attempt at a Solution

 
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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:

[tex] \frac{d^{2}y}{dx^{2}}+y=0[/tex]

Take the series representation for either sin or cos and see what happens.
 
oh i see now
just some simplification of the series combination
and it does cancel
Thank you
 
That's what I'm here for...
 

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