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Taken from Mathematica helps pages on DSolve
Solve for \(u(r,t)\) with \(0<r<1\) and \(t>0\).
\[ \frac {\partial ^2 u}{\partial t^2} = c^2 \left ( \frac {\partial ^2 u}{\partial r^2} + \frac {1}{r} \frac {\partial u}{\partial r} \right ) \]
With boundary conditions
\begin {align*} u(1,t) &=0 \end {align*}
With initial conditions
\begin {align*} u(r,0) &=1 \\ \frac {\partial u}{\partial t}(r,0) &= \frac {r}{3} \end {align*}
Mathematica ✓
ClearAll[u, t, r, n]; pde = D[u[r, t], {t, 2}] == c^2*(D[u[r, t], {r, 2}] + (1*D[u[r, t], r])/r); ic = {u[r, 0] == 1, Derivative[0, 1][u][r, 0] == r/3}; bc = u[1, t] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic, bc}, u[r, t], {r, t}], 60*10]]; sol = sol /. K[1] -> n; sol = FullSimplify[sol];
\[ \left \{\left \{u(r,t)\to \sum _{n=1}^{\infty }\frac {2 \text {BesselJ}(0,r \text {BesselJZero}(0,n)) \left (9 \sqrt {c^2} \text {BesselJ}(1,\text {BesselJZero}(0,n)) \cos (c t \text {BesselJZero}(0,n))+\text {HypergeometricPFQ}\left (\left \{\frac {3}{2}\right \},\left \{1,\frac {5}{2}\right \},-\frac {1}{4} \text {BesselJZero}(0,n)^2\right ) \sin \left (\sqrt {c^2} t \text {BesselJZero}(0,n)\right )\right )}{9 \sqrt {c^2} \left (\text {BesselJ}(0,\text {BesselJZero}(0,n))^2+\text {BesselJ}(1,\text {BesselJZero}(0,n))^2\right ) \text {BesselJZero}(0,n)}\right \}\right \} \]
Maple ✓
x:='x'; t:='t'; y:='y'; u:='u';c:='c'; pde := diff(u(r, t), t$2) = c^2*( diff(u(r,t), r$2)+ (1/r)* diff(u(r,t),r) ) ; ic:=u(r,0)=1, eval( diff(u(r,t),t),t=0)=r/3; bc:=u(1,t)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic,bc], u(r, t)) assuming t>0,r>0,r<1),output='realtime'));
\[ u \left ( r,t \right ) =-{\it invlaplace} \left ( {\frac {1}{s}\BesselI \left ( 0,{\frac {sr}{c}} \right ) \left ( \BesselI \left ( 0,{\frac {s}{c}} \right ) \right ) ^{-1}},s,t \right ) -1/3\,{\it invlaplace} \left ( {\frac {1}{{s}^{2}}\BesselI \left ( 0,{\frac {sr}{c}} \right ) \left ( \BesselI \left ( 0,{\frac {s}{c}} \right ) \right ) ^{-1}},s,t \right ) +1/6\,\pi \,c{\it invlaplace} \left ( {\frac {1}{{s}^{3}}\BesselI \left ( 0,{\frac {sr}{c}} \right ) \StruveL \left ( 0,{\frac {s}{c}} \right ) \left ( \BesselI \left ( 0,{\frac {s}{c}} \right ) \right ) ^{-1}},s,t \right ) -1/6\,\pi \,c{\it invlaplace} \left ( {\frac {1}{{s}^{3}}\StruveL \left ( 0,{\frac {sr}{c}} \right ) },s,t \right ) +1+1/3\,tr \] Has unresolved Invlaplace calls
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Solve for \(u(r,\theta ,t)\) with \(0<r<a\) and \(t>0\) and \(-\pi <\theta <\pi \)
\[ \frac {\partial ^2 u}{\partial t^2} = c^2 \left ( \frac {\partial ^2 u}{\partial r^2} + \frac {1}{r} \frac {\partial u}{\partial r} +\frac {1}{r^2} \frac {\partial ^2 u}{\partial \theta ^2} \right ) \]
With boundary conditions
\begin {align*} u(a,\theta ,t) &=0 \\ |u(0,\theta ,t)| < \infty \\ u(r,-\pi ,t) &= u(r,\pi ,t) \\ \frac {\partial u}{\partial \theta }(r,-\pi ,t) &= \frac {\partial u}{\partial \theta }(r,\pi ,t)\\ \end {align*}
With initial conditions
\begin {align*} u(r,\theta ,0) &= f(r,\theta ) \\ \frac {\partial u}{\partial t}(r,\theta ,0) &= 0 \end {align*}
Mathematica ✗
ClearAll[u, t, r, n, theta, a, f]; pde = D[u[r, theta, t], {t, 2}] == c^2*(D[u[r, theta, t], {r, 2}] + (1*D[u[r, theta, t], r])/r + (1*D[u[r, theta, t], {theta, 2}])/r^2); ic = {u[r, theta, 0] == f[r, theta], Derivative[0, 0, 1][u][r, theta, 0] == 0}; bc = {u[a, theta, t] == 0, u[r, -Pi, t] == u[r, Pi, t], Derivative[0, 1, 0][u][r, -Pi, t] == Derivative[0, 1, 0][u][r, Pi, t]}; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic, bc}, u[r, theta, t], {r, theta, t}, Assumptions -> {0 < r < a, a > 0, t > 0, -Pi < theta < Pi}], 60*10]];
\[ \text {Failed} \]
Maple ✗
x:='x'; t:='t'; y:='y'; u:='u';theta:='theta'; pde := diff(u(r, theta, t), t$2) = c^2*(diff(u(r, theta, t), r$2) + 1/r*diff(u(r, theta, t), r) + 1/r^2 *diff(u(r, theta, t), theta$2)); ic := u(r, theta, 0) = f(r, theta) , (D[3](u))(r, theta, 0) = 0; bc := u(a, theta, t) = 0, u(r, -Pi, t) = u(r, Pi, t), (D[2](u))(r, -Pi, t) = (D[2](u))(r, Pi, t); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic,bc], u(r, theta ,t),HINT = boundedseries(r=0))),output='realtime'));
\[ \text { sol=() } \]