Solve BVP using Eigenfunction Expansion

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The discussion focuses on solving the boundary value problem (BVP) y''[x] - y[x] = Cos[2x] using eigenfunction expansion, with boundary conditions y'[0] = y'[π] = 0. Participants question whether the corresponding Sturm-Liouville problem is f''[x] + λf = 0, noting that the negative sign in front of y[x] may affect the problem's formulation. The eigenfunction f[x] is identified as Cos[nx], and the approach involves calculating y[x] as a sum of coefficients multiplied by f[x]. However, discrepancies arise when plotting the Fourier expansion, leading to confusion about the correct eigenfunction expansion method. The conversation highlights the importance of correctly identifying the Sturm-Liouville problem to achieve accurate solutions.
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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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