I am solving the heat equation in spherical coordinates.
$$\frac{\ 1}{\alpha}\frac{\delta T}{\delta t}=\frac{1}{r^2}\frac{\delta}{\delta r}(r^2\frac{\delta T}{\delta r})=\frac{\delta^2 T}{\delta r^2}+ \frac{2}{r} \frac{\delta T}{\delta r}$$
Inicial condition CI: T(r,0)=Ti
Boundary condition 1, r=R: $$\frac{-k\delta T}{\delta r}=h(T(R,t)-Tinf)$$
Boundary condition 2, r=0:
$$\frac{\delta T}{\delta r}=0$$
$$T(r,t) = X(r)*G(t)$$
Substituting in the equation I got: $$\frac{\ 1}{\ R}(\frac{\delta^2 T}{\delta r^2}+ \frac{2}{r} \frac{\delta T}{\delta r})=\frac{\ 1}{\alpha G}\frac{\delta G}{\delta t}= -\lambda^2 $$
I already solved the temporal part: $$G(t)=A \ exp(-\lambda^2 \alpha t)$$
But I have problems to solve the spacial part:
$$R''+\frac{\ 2}{\ r}R'+R\lambda^2=0$$
I have been reading about it, but this part has been solved using the Euler-Cauchy equation and implying other assumptions which are not possible here.
How do I proceed? I'm just taking the differential equations course, so I'm not too much into the topic. I would thank so much your help.