Added May 30, 2019.
Airy PDE
Solve for \(u(x,t)\)
\[ u_t+u_{xxx} =0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, t], t] + D[u[x, t], {x,3}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to x^3 (c_{11} t+c_4)+x^2 (c_3-60 c_6 t)+x (c_2-24 c_5 t)-3 t (c_{11} t+2 c_4)-\frac {c_{11} x^6}{120}+c_6 x^5+c_5 x^4+c_1\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,t),t)+diff(u(x,t),x$3)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[u \left ( x,t \right ) ={\frac {{\it \_C4}}{{{\rm e}^{{\it \_c}_{{1}}t}}} \left ( {\it \_C1}\,{{\rm e}^{-{\frac {x}{2}\sqrt [3]{{\it \_c}_{{1}}}}}}{{\rm e}^{-{\frac {i}{2}}x\sqrt {3}\sqrt [3]{{\it \_c}_{{1}}}}}+{\it \_C2}\,{{\rm e}^{-{\frac {x}{2}\sqrt [3]{{\it \_c}_{{1}}}}}}{{\rm e}^{{\frac {i}{2}}x\sqrt {3}\sqrt [3]{{\it \_c}_{{1}}}}}+{\it \_C3}\,{{\rm e}^{\sqrt [3]{{\it \_c}_{{1}}}x}} \right ) }\]
Hand solution
Solve for \(u_{t}+u_{xxx}=0\) on the real line for \(t>0\). Let \(u=T\left ( t\right ) X\left ( t\right ) \). The pde becomes\begin {align*} T^{\prime }X+X^{\prime \prime \prime }T & =0\\ \frac {T^{\prime }}{T} & =-\frac {X^{\prime \prime \prime }}{X}=-\lambda \end {align*}
Hence \(X^{\prime \prime \prime }+\lambda X=0\). This ODE has solution \[ X\left ( x\right ) =C_{1}e^{\left ( -\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i}{2}\lambda ^{\frac {1}{3}}\sqrt {3}\right ) x}+C_{2}e^{\left ( -\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i}{2}\lambda ^{\frac {1}{3}}\sqrt {3}\right ) x}+C_{3}e^{\lambda ^{\frac {1}{3}x}}\] The ODE \(T^{\prime }+\lambda T=0\) has the solution \(T\left ( t\right ) =C_{4}e^{-\lambda t}\). Therefore the solution to the PDE is \(T\left ( t\right ) X\left ( t\right ) \) given by\[ u\left ( x,t\right ) =C_{4}e^{-\lambda t}\left ( C_{1}e^{\left ( -\frac {\lambda ^{\frac {1}{3}}}{2}-\frac {i}{2}\lambda ^{\frac {1}{3}}\sqrt {3}\right ) x}+C_{2}e^{\left ( -\frac {\lambda ^{\frac {1}{3}}}{2}+\frac {i}{2}\lambda ^{\frac {1}{3}}\sqrt {3}\right ) x}+C_{3}e^{\lambda ^{\frac {1}{3}x}}\right ) \]