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[tex]\frac{d^{2}\psi}{dx^{2}}[/tex] + ([tex]\lambda[/tex] - a[tex]^{2}[/tex]x[tex]^{2}[/tex])[tex]\psi[/tex] = 0

"a" and lambda are constants. Most books solve this by "assuming" that the solution is a product of a power series (polynomial) and a gaussian type function. Is there a more "rigorous" way to approach this problem without making such assumptions? Does this ODE have a name? I'd like to look more into it.

Thanks!