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- Homework Statement
- Solve for ##\frac{d^{2}\psi}{d\xi^2} =\xi^2\psi##

- Relevant Equations
- For unequal roots 2nd order ODE, the solution is: ##y=c_{1}e^{-kx}+c_{2}e^{kx}##

This is a very simple question: I would like to solve for ##\psi## in this equation $$\frac{d^{2}\psi}{d\xi^2} =\xi^2\psi$$

I so apply ##y=c_{1}e^{-kx}+c_{2}e^{kx}## and ##\psi## should be equal to ##\psi=c_{1}e^{-\xi^2}+c_{2}e^{\xi^2}##, because ##(D^2-\xi^2)\psi=0##. However the answer is ##\psi(\xi) = c_{1}e^{-\frac{\xi^2}{2}}+c_{2}e^{\frac{\xi^2}{2}}##. Why is it so?

I so apply ##y=c_{1}e^{-kx}+c_{2}e^{kx}## and ##\psi## should be equal to ##\psi=c_{1}e^{-\xi^2}+c_{2}e^{\xi^2}##, because ##(D^2-\xi^2)\psi=0##. However the answer is ##\psi(\xi) = c_{1}e^{-\frac{\xi^2}{2}}+c_{2}e^{\frac{\xi^2}{2}}##. Why is it so?