Solve BVP using Eigenfunction Expansion

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SUMMARY

The discussion focuses on solving the boundary value problem (BVP) defined by the equation y''[x] - y[x] = Cos[2x] using eigenfunction expansion methods. The boundary conditions specified are y'[0] = y'[π] = 0. The participants explore the implications of the Sturm-Liouville problem and question whether the negative sign in front of y[x] alters the corresponding eigenfunction expansion approach. The confusion arises when the Fourier expansion of y[x] does not match the expected results, indicating a potential misapplication of the Sturm-Liouville framework.

PREREQUISITES
  • Understanding of Sturm-Liouville theory
  • Familiarity with eigenfunction expansions
  • Knowledge of Fourier series and their applications
  • Basic differential equations, specifically second-order linear equations
NEXT STEPS
  • Study the properties of Sturm-Liouville problems in detail
  • Learn about eigenfunction expansions specific to non-standard boundary conditions
  • Review Fourier series convergence and its implications in solving BVPs
  • Explore the differences in eigenfunction expansions when negative coefficients are present
USEFUL FOR

Mathematics students, physicists, and engineers involved in solving differential equations and boundary value problems, particularly those utilizing eigenfunction expansions and Fourier series.

bitty
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Homework Statement


Solve BVP y''[x]-y[x]=Cos[2x] using eigenfunction expansion.
We know y'[0]=y'[Pi]=0

Homework Equations



Fourier it up

The Attempt at a Solution


Is the corresponding Sturm-Liouville problem:
f''[x]+/lambda*f=0?

All the examples we've done have been of form y''[x]+y[x]=g[x] but does a negative sign in front of y[x] change the corresponding S/L problem we use?

It follows that f[x]=Cosnx

Because I proceed to solve for y[x]=Sum[A_n*f[x]] and I find the coefficents and all but when I plot my Fourier expansion for y[x], it is very different from the actual result of y[x]. I'm guessing I wasn't supposed to use f''[x]+/lambda*f=0 to find my eigenfunction expansion, but I'm confused as to why I'm getting the wrong answer
 
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Edit: sorry i misunderstood
 

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