How to start an eigenvalue problem

Question

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems...

Please give me a hint or a clue, cause I don't know how to start. Thanks in advance.

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My first idea was to get basis like this $\phi_n(x)=x\left(1-x\right)^n $ and $$\phi(x)=\sum_{n=1}^{\infty} a_n \phi_n(x)$$ am I on a right track?

Answer 1

I guess DEQs of this type show up when expressing problems in terms of parabolic cylindrical coordinates.

So you might be looking for Weber DEQs or more generally, parabolic cylinder functions as a basis.

Fitting your problem into this framework, the solution could be

$$ \phi(\lambda, x) = c_1 D_{(-1/2)}((1+i) x^{1/4} \sqrt{x+2} \; \lambda)+c_2 D_{-1/2}((-1+i) x^{1/4} \sqrt{x+2} \; \lambda) $$

where $D_n (z)$ are parabolic cylinder functions and the constants should be adapted to the boundary conditions. Wolfram Alpha helps as well.

Hope this goes in the right direction...