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[tex]\frac{d^2\psi}{du^2}+(\frac{\beta}{\alpha}-u^2)\psi=0[/tex]

This is the SE for the simple harmonic oscillator. My text goes through an elaborate solution to this DE and ends up resorting to a power series solution, not for psi, but for H, where [tex]\psi=H(u)e^{-u^2/2}[/tex]. The text also points out that no power series solution could be found by directly substituting in the SE for psi. However, Apostols THM6.13 (Volume II) states that any 2nd order ODE with analytic coefficients has a power series solution. Which is right?

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# Power Series Solution (SE)?

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