Eigenfunction 2nd order DE problem

[JamesGoh]

Homework Statement

For the following equation

$y" - λy=0$

find the values of λ which produce a non-trivial solution on the interval 0 <= x <= a

The given initial conditions are

y(0) = 0
y(a) = 0

Homework Equations

The Attempt at a Solution

see attached pdfs

My problem is I am not being able to produce a non-trivial solution for all three cases. It's further complicated by the fact that the tutor hasn't provided any answer to the problem

I just want to make sure my understanding and approach to the problem is reasonable. If anyone has a good answer to the problem, please let me know

 

[LCKurtz]

It looks pretty much OK right up to the end. The equation is y''-λy = 0, y(0)=y(a) = 0.

You are correct that λ > 0 and λ = 0 yield no nontrivial solutions. In the case λ < 0 you can write λ = -k² so the equation as

y'' + k²y = 0

Then at the end when you get to Bsin(ka) = 0, to avoid B = 0 you must have
ka = nπ so k = nπ/a and λ_n = -(nπ/a)². The eigenvalues are always real in this type of problem.