How to solve this wave equation:
$ay^2\psi+by\psi+\frac {d^2\psi}{dx^2}=0$ here $c=\frac {d^2\psi}{dx^2}$ where, the equation is a quadratic equation (a univariate polynomial equation of the second degree). example: $ay^2+by+c=0,$
$y=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$
the problem is that here there are two wave functions after $y^2$ and $y$:
$ay^2\psi+by\psi+c=0,$
thats why i dont know how to solve it.
Remark: Is this solution true?
$(y+\frac {b}{2a})^2\psi=\frac {(b^2\psi-4ac)}{4a^2},$ to solve this i take the wave function under the radical. $y\sqrt {\psi}=\frac{-b\sqrt {\psi}\pm\sqrt{b^2\psi-4ac}}{2a}$
isn't it wrong to take take the wave function under the radical$\sqrt {\psi}$ ?