I'm given the following differential equation
$$\frac{\partial^2 E}{\partial z^2}-\frac{1}{u_p^2}\frac{\partial^2 E}{\partial t^2}=0$$
and asked the following:
Show that any function in the form of $F(z-u_pt)+G(z+u_pt)$ will satisfy this differential equation ($F$ and $G$ are arbitrary functions).
Now, I'm not quite sure how I'm supposed to take a partial derivative of an arbitrary function. I have some work, and ended up with the right answer, but I'm almost positive that this cannot be correct. Can someone. please point me in the right direction?
My work:
I realize now as I upload my pictures that choosing a function $U$ was a poor decision. $U_p$ and $U(z,t)$ are not the same (Sorry).
Best Answer
Congratulations. You have made use of chain rule correctly provided those partial derivatives exist.