Confusion with changing variables in second order DE So in my physics assignment, we're given Schrodinger's equation:
$$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2}+e\xi x\psi=E\psi$$
We're asked to substitute a function of the form $w(x)=Ax+B$ to arrive at an equation of the form
$$\frac{\partial^2\psi}{\partial w^2}-w\psi=0$$
So I have a fair idea of how to approach this, and I've made good progress, but I have couple points of confusion:


*

*Where does the partial come from? Since the function is explicitly and implicitly of one variable, shouldn't we simply have full derivatives?

*Am I using the chain rule correctly here:
$$\frac{d\psi}{dw}=\frac{d\psi}{dx}\frac{dx}{dw}$$
$$\frac{d^2\psi}{dw^2}=\frac{d}{dw}\left(\frac{d\psi}{dx}\frac{dx}{dw}\right)=\frac{d^2\psi}{dx^2}\frac{dx}{dw}+\frac{d\psi}{dx}\frac{d^2x}{dw^2}$$


Answer:
1. Physics lecturers abuse notation, so here partials are used interchangeably with full derivatives.
2.
$$\frac{d}{dw}\left(\frac{d\psi}{dx}\frac{dx}{dw}\right)=\frac{d}{dw}\frac{d\psi}{dx}\frac{dx}{dw}+\frac{d\psi}{dx}\frac{d}{dw}\frac{dx}{dw}=\frac{dx}{dw}\frac{d}{dx}\frac{d\psi}{dx}\frac{dx}{dw}+\frac{d\psi}{dx}\frac{d^2x}{dw^2}=\frac{d^2\psi}{dx^2}\left(\frac{dx}{dw}\right)^2+\frac{d\psi}{dx}\frac{d^2x}{dw^2}$$
