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**1. The problem statement, all variables and given/known data**

Find all eigenvalues and eigenfunctions:

[itex] -y'' (x) = \lambda y(x) , x \in (a,b) [/itex]

[itex] y(a)= y(b) =0 [/itex].

**2. Relevant equations**

[itex] sin x = \frac{e^{ix} + e^{-ix} }{2i} [/itex].

**3. 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!

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**