I'm trying to solve Laplace's equation in the disc $r\leq a$ using separation of variables (so relatively simple stuff, compared to what's often on here), and I've proceeded like so thus far:

Let $T(r,\theta)$ be a solution to $\frac{\partial^2 T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{1}{r^2}\frac{\partial^2 T}{\partial \theta^2}=0$.
Substituting $T(r,\theta)=F(r)G(\theta)$ into this equation, and equating both functions to a constant, say, $\lambda$, gives:

(1) $r^2F''(r)+rF'(r)=\lambda F(r)$

(2) $G''(\theta)=-\lambda G(\theta)$

First considering (2), we know that $G(\theta)$ must be $2\pi$ periodic, which is only possible if either $\lambda=n^2$ for $n\in\mathbb{N}$, in which case $G(\theta)=A\sin(n\theta)+B\cos(n\theta)$, or $\lambda=0$, giving $G(\theta)=c$ for some constant $c$.

Now looking back at (1), $\lambda=0$ gives $F(r)=A + B\log{r}$, but what about when $\lambda=n^2$?

I have in my online notes that the solution is a linear combination of $r^n$ and $r^{-n}$, and I can see that these fit, but apart from getting that answer by a lucky guess, I don't see how one would come about it. The notes say that this equation is of 'Euler's type', but I'm not too sure what that means either, and if I search for it on the internet I just come across a lot of other unrelated equations that Euler had something to do with.

Thanks in advance, and this is my first question, so sorry if I've done anything wrong!