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[tex] \frac{-\hbar^2}{2m}\frac{d^2}{dx^2}u(x) + \frac{1}{2}m\omega^2 x^2 u(x) = Eu(x)[/tex]

Into its dimensionless form:

[tex]\frac{d^2}{dy^2}u(y) + (2\epsilon - y^2)u(y) = 0[/tex]

I have the following info:

[tex]E = \epsilon\hbar\omega[/tex]

[tex]x = y\sqrt{\frac{\hbar}{m\omega}}[/tex]

Heres what ive done so far:

[tex] \frac{-\hbar^2}{2m}\frac{d^2}{dx^2}u(x) + \frac{1}{2}m\omega^2 y^2 \frac{\hbar}{m\omega} u(x) = \epsilon\hbar\omega u(x)[/tex]

[tex] \frac{-\hbar}{m}\frac{d^2}{dx^2}u(x) + \omega^2 y^2 u(x) = 2\epsilon\omega u(x)[/tex]

[tex] \frac{\hbar}{m}\frac{d^2}{dx^2}u(x) + 2\epsilon\omega u(x) - \omega^2 y^2 u(x) = 0[/tex]

[tex] \frac{\hbar}{m}\frac{d^2}{dx^2}u(x) + (2\epsilon - y^2)\omega u(x) = 0[/tex]

But i cant see where to go next..i know i must be close to the end though..surely!