1. ## Eigenfunctions and Eigenvalues

Find the eigenfunctions and eigenvalues of the equation $\phi '' - 2 \phi ' + \lambda \phi = 0$ with boundary conditions $\phi(0) = 0$ and $\phi(L) = 0$

2. Is $\lambda$ a constant or a function? What is $\phi$ a function of?

3. This is a Sturm-Liouville Problem.

The aux equation is $r^2 - 2r + \lambda = 0$

$r = \frac{2 \pm \sqrt{4 - 4 \lambda}}{2}$

I have found that for $4 - 4 \lambda = 0$ and $4 - 4 \lambda > 0$ the solutions are trivial.

For $4 - 4 \lambda < 0$ though

we get complex roots $r = 1 \pm \ i \frac{\sqrt{4-4 \lambda}}{2}$

so the ODE solution is

$\phi = c_1 e^x \text{sin}(\frac{\sqrt{4 \lambda - 4}}{2}x) + c_2 e^x \text{cos} (\frac{\sqrt{4 \lambda - 4}}{2}x)$

Plugging in the first boundary condition we will get

$\phi(0) = c_2 = 0$ so $c_2 = 0$

Plugging in the second boundary condition we will get

$\phi(L) = c_1 e^L \text{sin}(\frac{\sqrt{4 \lambda - 4}}{2}L) = 0$

Now
For a non-trivial solution we assume the $c_2 \neq 0$ which implies

$\text{sin}(\frac{\sqrt{4 \lambda - 4}}{2}L) = 0$

Since zeros of Sin only occur at integral multiples of $\pi$, to satisfy the boundary condition we must have

$\frac{\sqrt {4 \lambda-4}}{2}L= n \pi$ for $n = 1,2....$

Then $\lambda_n = \frac{L^2 + (n \pi)^2}{L^2}$ is the eigenvalue, but Im having trouble with the eigenfunction......

4. You have done the hard part, which is to find the eigenvalues, namely $\lambda_n = 1 + \bigl(\tfrac{n\pi}L\bigr)^2$ (though you could have made the calculation a bit simpler if you had noticed that $\tfrac{\sqrt{4\lambda-4}}2 = \sqrt{\lambda-1}$).
For the eigenfunctions, just substitute the value $\lambda = \lambda_n$ into the equation $\phi(x) = c_1 e^x \sin(\sqrt{ \lambda - 1}\,x)$ to get $\boxed{\phi_n(x) = c_1 e^x \sin\bigl(\tfrac{n\pi x}L\bigr)}$.