General Solution to the Radial Component Equation in the Laplace's Equation in Spherical Coordinates
Theorem
The general solution of the radial part equation in the spherical coordinate system Laplace’s equation is given below.
$$ R(r)=\sum \limits_{l=0}^{\infty}R_{l}(r)=\sum \limits_{l=0}^{\infty}\left( A_{l}r^{l}+\frac{ B_{l}}{r^{l+1}} \right) $$
Here, $l$ is a non-negative integer, and $A_{l}$, $B_{l}$ are constants.
Description
The process of finding this is relatively simple compared to the solution for polar and azimuthal angles.
Proof
In the spherical coordinate system Laplace’s equation, the solutions for the polar angle $\theta$ and azimuthal angle $\phi$ are called spherical harmonics. In the process of finding the spherical harmonics, we obtain the equation for the radial component as follows.
$$ \frac{1}{R}\frac{ d }{ dr }\left( r^{2}\frac{ d R}{ dr } \right)=l(l+1) $$
Here, $l$ is a non-negative integer. Reorganizing this equation yields the following.
$$ r^{2}\frac{ d^{2} R}{ dr^{2} }+2r\frac{ d R}{ dr }-l(l+1)R=0 $$
Such a form of differential equation is known as an Euler differential equation. It is well known that the solution to the Euler differential equation is in the form of $R(r)=r^{k}$. Substituting this results in the following.
$$ \begin{align*} && k(k-1)r^{k}+2kr^{k}-l(l+1)r^{k}=0 \\ \implies && [k^{2}+k-l(l+1)]r^{k}=0 \\ \implies && k^{2}+k-l(l+1)=0 \end{align*} $$
This is a simple quadratic equation. Solving it gives $k_{1}=l$, $k_{2}=-l-1$. Thus, we obtain the following.
$$ R(r)=\sum \limits_{l=0}^{\infty}R_{l}(r)=\sum \limits_{l=0}^{\infty}\left( A_{l}r^{l}+\frac{ B_{l}}{r^{l+1}} \right) $$
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