Sturm-Liouville is a special type of boundary value problem.
First consider with not equal zero:
The auxiliary equation has roots .
Then the solution is of the form
Now use the boundary conditions to solve for the constants and .
From the boundary conditions y'(0) = 0 and y(1) + y'(1) = 0 you can establish the following system of linear equations:
3. Make sure you use the non-trivial solution to ... I will leave it to you to convince yourself that non-trivial solutions only exist for .
The solutions will be in the form of trigonometric functions because the characteristic equation has complex roots.