2.9.1 $u_{xx}+u_{yy}+\frac{\beta }{x}{u}_x=0$

problem number 95

Added June 20, 2019

Taken from http://people.maths.ox.ac.uk/chengq/outreach/The%20Tricomi%20Equation.pdf

Solve for $u(x,y)$ $$u_{xx}+u_{yy}+\frac{\beta }{x}{u}_x=0$$

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] + beta/x*D[u[x,y],x] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}, Assumptions->beta>0], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(u(x,y),x$2)+ diff(u(x,y),y$2) + beta/x*diff(u(x,y),x)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y),'build') assuming beta>0),output='realtime'));
 

$$u(x,y)=(c_1\mathrm{BesselJ}(\frac{\beta }{2}-\frac{1}{2},\sqrt{-{\mathit{\text{\_c}}}_1}x)+c_2\mathrm{BesselY}(\frac{\beta }{2}-\frac{1}{2},\sqrt{-{\mathit{\text{\_c}}}_1}x))(c_3\sin \left(y\sqrt{{\mathit{\text{\_c}}}_1}\right)+c_4\cos \left(y\sqrt{{\mathit{\text{\_c}}}_1}\right))\sqrt{x}{x}^{-\frac{\beta }{2}}$$

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