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Eigenfunction expansion

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Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution.


I use the function


to get the eigenfunction is

ϕ=Acosxλ+BcosxλϕAxλBxλ\phi=A\cos x\sqrt{\lambda}+B\cos x\sqrt{\lambda}

but how should I decide A and B? Is it by system


or it should be


and why? After getting eigenvalue and eigenfunctions, what should I do? I hope somebody can give me a answer with details. Thanks.

i would say ϕ′⁢(0)=α,ϕ′⁢(1)=β But you made a mistake in that it should be

Asin(... +Bcos(.... ANd not both cos(....

And then you should be able to solve A,B for arbitrary a and beta as long as lambda is not an integer multiple of pi^2 in which case there may not be a solution - also you should have chosen different symbols because a and A could be taken to be the same when they need to be different arbitrary and likewise beta and B.

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