I am reviewing some quantum mechanics and and have come across a solution to a differential equation that I do not understand in the derivation of the quantum harmonic oscillator. The Schrodinger equation for a 1D harmonic oscillator is:
$$-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\psi=E\psi$$
with the introduction of:
$$K=\frac{2E}{\hbar\omega}, \xi=\sqrt{\frac{m\omega}{\hbar}}x$$
for $\xi>>K$, we get:
$$\frac{d^2 \psi }{d\xi ^2} \approx \xi^2\psi$$
With solution:
$$\psi = A\exp\left(-\dfrac{\xi^2}{2}\right) + B\exp\left(\dfrac{\xi^2}{2}\right )$$
Where I was expecting the solution to be:
$$\psi = A\exp(-{\xi}) + B\exp({\xi}).$$
Can someone point out what I am missing? Thanks in advance