Taken from Maple pdsolve help pages
Solve for \(u( r,\theta ,\phi )\). Where \(\theta \) is the polar angle and \(\phi \) is the azimuthal angle. Hence \(0<\theta <\pi \) and \(-\pi <\phi <\pi \).
\begin {align*} \frac {\partial }{\partial r} \left (r^2 \frac {\partial u}{\partial r} \right ) + \frac {1}{\sin \theta } \frac {\partial }{\partial \theta } \left (\sin \theta \frac {\partial u}{\partial \theta } \right ) + \frac {1}{\sin ^2\theta } \frac {\partial ^2 u}{\partial \phi ^2}=0 \end {align*}
Mathematica ✓
ClearAll["Global`*"]; lap = Laplacian[f[r, theta, phi], {r, theta, phi}, "Spherical"]; sol = AbsoluteTiming[TimeConstrained[DSolve[lap == 0, f[r, theta, phi], {r, theta, phi}], 60*10]];
\[\left \{\left \{f(r,\theta ,\phi )\to \begin {array}{cc} \{ & \begin {array}{cc} \sqrt {2} r^{-\frac {1}{2} \sqrt {4 c_7+1}-\frac {1}{2}} \left (c_1 r^{\sqrt {4 c_7+1}}+c_2\right ) \left (c_4 \, _2F_1\left (\frac {1}{4} \left (-\sqrt {4 c_7+1}+2 \sqrt {c_8}+1\right ),\frac {1}{4} \left (\sqrt {4 c_7+1}+2 \sqrt {c_8}+1\right );\frac {1}{2};\cos ^2(\theta )\right )+c_3 \cos (\theta ) \, _2F_1\left (\frac {1}{4} \left (-\sqrt {4 c_7+1}+2 \sqrt {c_8}+3\right ),\frac {1}{4} \left (\sqrt {4 c_7+1}+2 \sqrt {c_8}+3\right );\frac {3}{2};\cos ^2(\theta )\right ) \text {sgn}(\sin (\theta ))\right ) \left (-\sin ^2(\theta )\right )^{\frac {\sqrt {c_8}}{2}} \left (c_6 \cos \left (\phi \sqrt {c_8}\right )+c_5 \sin \left (\phi \sqrt {c_8}\right )\right ) & -\pi \leq \theta \leq \pi \land 0\leq \phi \leq \pi \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart; PDE := diff(r^2*diff(F(r,theta,phi),r),r)+ 1/sin(theta)*diff(sin(theta)*diff(F(r,theta,phi),theta),theta)+ 1/sin(theta)^2*diff(F(r,theta,phi),phi$2) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(PDE,F(r,theta,phi),'build')),output='realtime'));
\[F \left ( r,\theta ,\phi \right ) ={ \left ( {\it \_C5}\,\sin \left ( \sqrt {{\it \_c}_{{2}}}\phi \right ) +{\it \_C6}\,\cos \left ( \sqrt {{\it \_c}_{{2}}}\phi \right ) \right ) \left ( {\it \_C1}\,{r}^{{\frac {1}{2}\sqrt {1+4\,{\it \_c}_{{1}}}}}+{\it \_C2}\,{r}^{-{\frac {1}{2}\sqrt {1+4\,{\it \_c}_{{1}}}}} \right ) \left ( \sqrt {2}\cos \left ( \theta \right ) {\it csgn} \left ( \cos \left ( \theta \right ) \right ) \hypergeom \left ( [{\frac {1}{2}\sqrt {{\it \_c}_{{2}}}}+{\frac {1}{4}\sqrt {1+4\,{\it \_c}_{{1}}}}+{\frac {3}{4}},{\frac {1}{2}\sqrt {{\it \_c}_{{2}}}}-{\frac {1}{4}\sqrt {1+4\,{\it \_c}_{{1}}}}+{\frac {3}{4}}],[{\frac {3}{2}}],{\frac {\cos \left ( 2\,\theta \right ) }{2}}+{\frac {1}{2}} \right ) {\it \_C4}+{\it \_C3}\,\hypergeom \left ( [{\frac {1}{2}\sqrt {{\it \_c}_{{2}}}}+{\frac {1}{4}\sqrt {1+4\,{\it \_c}_{{1}}}}+{\frac {1}{4}},{\frac {1}{2}\sqrt {{\it \_c}_{{2}}}}-{\frac {1}{4}\sqrt {1+4\,{\it \_c}_{{1}}}}+{\frac {1}{4}}],[{\frac {1}{2}}],{\frac {\cos \left ( 2\,\theta \right ) }{2}}+{\frac {1}{2}} \right ) \right ) \left ( - \left ( \sin \left ( \theta \right ) \right ) ^{2} \right ) ^{{\frac {1}{2}\sqrt {{\it \_c}_{{2}}}}}{\frac {1}{\sqrt {r}}}}\]
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