Boundary conditions for shooting method on quantum harmonic oscillator ODE

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mistergrinch
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Here's a simple numerical analysis problem that is confusing me. Can someone help me understand what boundary conditions to use here?

f''(x) - (x^2 - E_n) * f(x) = 0;

Assume f -> 0 as x -> +- inf. This equation comes from Schrödinger's equation for a one dimensional trapping potential, with E_n proportional to energy.

I am supposed to find the first five eigenvalues and eigenvectors with a shooting method, using x in [-4,4], and normalizing f so that int(f^2) = 1;

I'm not given any boundary conditions, so I'm not sure how to solve this problem. Can anyone help me understand what is going on here? Thanks!
 

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If you are asked to find eigenvalues, then your boudary conditions have to be y(-4)= y(4)= 0.