In the spherical coordinates \(\rho, \theta, \phi(\rho>0,0 \leq \theta<2 \pi, 0
\leq \phi \leq \pi)\) defined by the equations
$$
x=\rho \cos \theta \sin \phi, \quad y=\rho \sin \theta \sin \phi, \quad z=\rho
\cos \phi
$$
Laplace's equation is
$$
\rho^{2} u_{\rho \rho}+2 \rho u_{\rho}+\left(\csc ^{2} \phi\right) u_{\theta
\theta}+u_{\phi \phi}+(\cot \phi) u_{\phi}=0
$$
(a) Show that if \(u(\rho, \theta, \phi)=\mathrm{P}(\rho) \Theta(\theta)
\Phi(\phi),\) then \(\mathrm{P}, \Theta,\) and \(\Phi\) satisfy ordinary
differential equations of the form
$$
\begin{aligned} \rho^{2} \mathrm{P}^{\prime \prime}+2 \rho
\mathrm{P}^{\prime}-\mu^{2} \mathrm{P} &=0 \\ \Theta^{\prime
\prime}+\lambda^{2} \Theta &=0 \\\\\left(\sin ^{2} \phi\right) \Phi^{\prime
\prime}+(\sin \phi \cos \phi) \Phi^{\prime}+\left(\mu^{2} \sin ^{2}
\phi-\lambda^{2}\right) \Phi &=0 \end{aligned}
$$
The first of these equations is of the Euler type, while the third is related
to Legendre's equation.
(b) Show that if \(u(\rho, \theta, \phi)\) is independent of \(\theta,\) then the
first equation in part (a) is unchanged, the second is omitted, and the third
becomes
$$
\left(\sin ^{2} \phi\right) \Phi^{\prime \prime}+(\sin \phi \cos \phi)
\Phi^{\prime}+\left(\mu^{2} \sin ^{2} \phi\right) \Phi=0
$$
(c) Show that if a new independent variable is defined by \(s=\cos \phi\), then
the equation for \(\Phi\) in part (b) becomes
$$
\left(1-s^{2}\right) \frac{d^{2} \Phi}{d s^{2}}-2 s \frac{d \Phi}{d s}+\mu^{2}
\Phi=0, \quad-1 \leq s \leq 1
$$
Note that this is Legendre's equation.
