I'm a bit confused as to how my textbook has taken on this question:
By using the trial function $y=x^n$, find the general solution to the differential equation $$x^2\frac{d^2y}{dx}+x\frac{dy}{dx}-9y\:=\:0\:$$
I've done most of it and have come down to n = $3$ or $-3$. But how do I form the general solution from here? All I know is that the solutions to the auxiliary equation are $3$ and $-3$. My textbook says that the general solution is given by $$Ax^3 + Bx^{-3}$$
How did they get there, and why is there no $e$?
Right, so you have already done the difficult bit. You have correctly concluded that $y=x^n$ solves the differential equation if $n=3$ or $n=-3$. Next, we just observe that the differential equation is linear, meaning that if $y_1(x)$ and $y_2(x)$ solve the differential equation, then so does $y(x)=Ay_1(x)+By_2(x)$ for any value of $A$ or $B$. That is why the general solution is $y(x)=Ax^3+Bx^{-3}$. There is no $e$ because we do not need it, not all differential equations need it.
I hope that helps.