Solve Eigenfunctions & Eigenvalues w/Ly=-y'', Boundary Conditions

[faiza muneer]

Homework Statement

Find eigenfunctions and eigenvalues to the operator
Ly=-y''
boundary conditions are y(0)=y'(1)=0. express f(x)=x²-2x as aseries of the eigenfunctions. use this to calculate $$\sum$$$$\frac{1}{(2k+1)⁶}$$

Homework Equations

The Attempt at a Solution

i attempted as shown in attached files but did not get the right answer can anyone help me

 

[vela]

Your results, so far, look okay to me. You need to decide on whether your index variable is going to be called n or k. Stick with just one. Also, the summation should start from n=0, not n=1.