Everything looks great so far!
The fact that $\Theta(\theta)$ is periodic with period $2\pi$ tells you that $\sqrt{k}$ must be a non-negative integer:
$$\sqrt{k} = n , \ \ \ \ \ \ \ \ \ \ n\in \{ 0, 1, 2, 3, \dots \} $$
That's all!
Having observed this, you now need to solve the equation for $R(r)$:
$$ r^2 R''(r) + r R'(r) = n^2 R(r).$$
I'll leave you to verify that the solutions are
$$ R(r) = \begin{cases} C + D \ln r & n = 0, \\ Cr^n + Dr^{-n} & n = 1,2,3,\dots \end{cases} $$
[If you want to derive this from scratch, you could observe that the ODE is homogenous, which means that it is a good idea to try the substitution $r = e^t$.]
Now try to use the fact that you're working in the region $r < 1$ to rule out $\ln r$ and $r^{-n}$ as invalid solutions.
Finally take a linear superposition of all the solutions that you have found to obtain the most general solution. Good luck!
Let's summarize the derivation in Whitham's Linear and Nonlinear Waves, $\S 4.1$: The Cole-Hopf Transformation
Burgers' equation
The Burgers equation is the simplest example of a partial differential equation demonstrating both diffusive and nonlinear propagation effects.
$$
u_{\color{red}{t}} + u u_{\color{blue}{x}} = \nu u_{\color{blue}{xx}}
\tag{1}
$$
Coloring distinguishes $\color{red}{time}$ derivatives from $\color{blue}{space}$ derivatives.
Cole-Hopf transform
The nonlinear transform of Cole and Hopf is,
$$
u = -2\nu \frac{\varphi_{\color{blue}{x}}}{\varphi},
$$
similar to the Thomas transformation of exchange equations. The transformation is resolved into two steps.
Step 1
Set $$u = \psi_{\color{blue}{x}},$$ and equation (1) becomes
$$
\psi_{\color{red}{t}} + \frac{1}{2}\left(\psi_{\color{blue}{x}}\right)^{2} = \nu \psi_{\color{blue}{xx}}
\tag{2}
$$
Step 2
Set $$\psi = -2\nu \ln \varphi,$$ and equation (2) reduces to the heat equation
$$
\boxed{
\varphi_{\color{red}{t}} = \nu \varphi_{\color{blue}{xx}}
}
\tag{3}
$$
The nonlinear term has vanished because of the nonlinear transformation.
Transforming heat equation solutions to Burgers equation solution
The initial value problem which starts with a spatial waveform
$$
u = F(x), \qquad t =0
$$
transforms into
$$
\varphi = \Phi(x) = e^{-\frac{1}{2\nu}\int_{0}^{x} F(\varphi)d\eta}, \qquad t = 0
$$
The heat equation solution for this IVP is
$$
\varphi = \frac{1}{\sqrt{4\pi \nu t}} \int_{-\infty}^{\infty} \Phi(\eta)e^{-\frac{(x-\eta)^{2}}{4\nu t}} d\eta
$$
Going back to the Cole-Hopf transform to recover the solution function $u$:
$$
u(x,t) = \frac
{\int_{-\infty}^{\infty} \frac{x-\eta}{t} e^{-\frac{G}{2\nu}} d\eta}
{\int_{-\infty}^{\infty} e^{-\frac{G}{2\nu}} d\eta}
$$
with the function
$$
G\left(\eta; x, t \right) =\frac{\left( x - \eta \right)^{2}}{2t} +
\int_{\eta}^{0} F\left(\xi \right) d\xi
$$
Best Answer
One often divides PDEs into three main types, depending essentially on the order of the derivatives involved, and the sign in front of those derivatives:
Elliptic: all derivatives are second-order, all signs are positive. Example: $\partial^2_{xx} + \partial^2_{yy} + \partial^2_{zz}$, Laplace's equation.
Hyperbolic: all derivatives are second-order, one sign is negative, the rest are positive. Example: $-\partial^2_{tt} + \partial^2_{xx} + \partial^2_{yy}$, the wave equation.
Parabolic: one derivative is first order, the rest are second order, one sign is negative (corresponding to the "one derivative" term), the rest are positive. Example $-\partial_{t} + \partial^2_{xx} + \partial^2_{yy}$, the heat equation.
This is, of course, greatly oversimplifying things, but we tend to break our equations up into these categories because, within a particular category, similar characteristics are exhibited. For instance, parabolic equations smooth data instantly (even if the initial data for a heat equation has discontinuities, for any time $t > 0$, the solution is smooth), hyperbolic equations on the other hand propagate their singularities in a predictable way (along rays).