Finding Eigenvalues and Eigenfunctions for O.D.E. Problem on Interval [0, 4π]

[jegues]

Homework Statement

Solve the differential equation eigenvalue problem:

$$f'' + \lambda f = 0, \quad 0 \leq x \leq 4\pi, \quad \text{where} \quad f^{'}(0) =0, \quad f^{'}(4\pi) = 0, \quad \text{and} f \neq 0.$$

Consider ONLY $$\quad \lambda \geq 0, \quad$$ and find the values of $$\quad \lambda \quad$$ and f(x).

Homework Equations

The Attempt at a Solution

See figure attached for my attempt at the solution

By "Solve the differential equation eigenvalue problem" do they simply mean find all the eigen values and eigen functions?

If so, is what I've done correct?

Thanks again!

 

[LCKurtz]

It looks OK to me.

For extra credit, instead of just assuming λ ≥ 0, try λ = -μ² < 0 and show you only get the trivial solution.