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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