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eigenfunction expansion to represent the best solution

Mathematics Asked by Rubidium. on December 17, 2020

Use the appropriate eigenfunction expansion (if it exists) to represent the best solution of the following problems.

$$u”+ u =f(x)$$ with boundary condition $$u(0) = u(2pi), u'(0) = u'(2pi)$$

What I have now is trying to solve $$u”+u=lambda u$$
So the general equation is $$Acos(sqrt{1-lambda}x)+Bsin(sqrt{1-lambda}x)$$
Then using the boundary conditions to set a matrix, the eigenfunctions exists if only if $$cos(2pisqrt{1-lambda})=1$$Therefore $$2pisqrt{1-lambda} =2pi n $$So $$lambda = 1-n^2$$
and eigenfunction would be $$u_n = a_nsin nx+b_ncos nx$$

Am I on the right track ?
Is this simply the Fourier expansion ?

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