- #1
Combinatorics
- 36
- 5
Homework Statement
Find all eigenvalues and eigenfunctions:
[itex] -y'' (x) = \lambda y(x) , x \in (a,b) [/itex]
[itex] y(a)= y(b) =0 [/itex].
Homework Equations
[itex] sin x = \frac{e^{ix} + e^{-ix} }{2i} [/itex].
The Attempt at a Solution
So actually the only problem I have is to find the eigenfunctions (which should be something like [itex] sin \{ \frac{n \pi (x-a) }{(b-a)} \} [/itex] ) .
I received the eigenvalues are: [itex] \lambda_n = \frac{n^2 \pi^2 }{(b-a)^2} [/itex].
But how can I solve the two equations system I receive when applying these eigenvalues to the solution: [itex] y= c_1 e^{i \sqrt{\lambda_n} x} + c_2 e^{-i \sqrt{\lambda_n} x} [/itex].
I hope someone will be able to help me solve this two equations-system.
Thanks in advance!