Mathematica ✓
ClearAll["Global`*"];
lap = Laplacian[u[r, theta, phi, t], {r, theta, phi}, "Spherical"];
pde = D[u[r, theta, phi, t], {t, 2}] == c^2*lap;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[r, theta, phi, t], {r, theta, phi, t}, Assumptions -> {0 < theta < Pi}], 60*10]];
\[\left \{\left \{u(r,\theta ,\phi ,t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {\sqrt {2} e^{-\frac {1}{2} \sqrt {c_{10}} (2 \phi +\pi )-t \sqrt {c_{11}}} \left (\operatorname {BesselJ}\left (\frac {1}{2} \sqrt {\frac {4 c_9}{c^2}+1},\frac {r \sqrt {-c_{11}}}{\sqrt {c^2}}\right ) c_1+\operatorname {BesselY}\left (\frac {1}{2} \sqrt {\frac {4 c_9}{c^2}+1},\frac {r \sqrt {-c_{11}}}{\sqrt {c^2}}\right ) c_2\right ) \left (e^{2 \phi \sqrt {c_{10}}} c_5+c_6\right ) \left (e^{2 t \sqrt {c_{11}}} c_7+c_8\right ) \left (c_4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+1\right ),\frac {1}{4} \left (\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+1\right ),\frac {1}{2},\cos ^2(\theta )\right )+c_3 \cos (\theta ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+3\right ),\frac {1}{4} \left (\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+3\right ),\frac {3}{2},\cos ^2(\theta )\right )\right ) \sin ^{i \sqrt {c_{10}}}(\theta )}{\sqrt {r}} & c\neq 0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart;
lap:=VectorCalculus:-Laplacian( u(r,theta,phi,t), 'spherical'[r,theta,phi] );
pde := diff(u(r,theta,phi,t),t$2)= c^2* lap;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(r,theta,phi,t),'build') assuming 0<theta,theta<Pi),output='realtime'));
sol := simplify(sol);
\[u \left (r , \theta , \phi , t\right ) = \frac {{\mathrm e}^{\frac {\left (-\pi -2 \phi \right ) \sqrt {\textit {\_c}_{3}}}{2}-\sqrt {\textit {\_c}_{4}}\, t} \sin \left (\theta \right )^{i \sqrt {\textit {\_c}_{3}}} \left ({\mathrm e}^{2 \sqrt {\textit {\_c}_{4}}\, t} c_{7} +c_{8} \right ) \left ({\mathrm e}^{2 \sqrt {\textit {\_c}_{3}}\, \phi } c_{5} +c_{6} \right ) \left (\operatorname {BesselJ}\left (\frac {\sqrt {\frac {c^{2}+4 \textit {\_c}_{1}}{c^{2}}}}{2}, \frac {\sqrt {-\textit {\_c}_{4}}\, r}{c}\right ) c_{1} +\operatorname {BesselY}\left (\frac {\sqrt {\frac {c^{2}+4 \textit {\_c}_{1}}{c^{2}}}}{2}, \frac {\sqrt {-\textit {\_c}_{4}}\, r}{c}\right ) c_{2} \right ) \left (\operatorname {hypergeom}\left (\left [\frac {2 \sqrt {-\textit {\_c}_{3}}\, c -\sqrt {c^{2}+4 \textit {\_c}_{1}}+3 c}{4 c}, \frac {2 \sqrt {-\textit {\_c}_{3}}\, c +\sqrt {c^{2}+4 \textit {\_c}_{1}}+3 c}{4 c}\right ], \left [\frac {3}{2}\right ], \cos \left (\theta \right )^{2}\right ) \cos \left (\theta \right ) c_{4} +\operatorname {hypergeom}\left (\left [\frac {2 \sqrt {-\textit {\_c}_{3}}\, c -\sqrt {c^{2}+4 \textit {\_c}_{1}}+c}{4 c}, \frac {2 \sqrt {-\textit {\_c}_{3}}\, c +\sqrt {c^{2}+4 \textit {\_c}_{1}}+c}{4 c}\right ], \left [\frac {1}{2}\right ], \cos \left (\theta \right )^{2}\right ) c_{3} \right )}{\sqrt {r}}\]
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