2.10.1 \(u_{xx} + u_{yy} + \frac {\beta }{x} u_x = 0\)

problem number 102

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'));
 

\[u \left (x , y\right ) = x^{\frac {1}{2}-\frac {\beta }{2}} \left (c_{3} \sin \left (\sqrt {\textit {\_c}_{1}}\, y \right )+c_{4} \cos \left (\sqrt {\textit {\_c}_{1}}\, y \right )\right ) \left (\operatorname {BesselJ}\left (\frac {\beta }{2}-\frac {1}{2}, \sqrt {-\textit {\_c}_{1}}\, x \right ) c_{1} +\operatorname {BesselY}\left (\frac {\beta }{2}-\frac {1}{2}, \sqrt {-\textit {\_c}_{1}}\, x \right ) c_{2} \right )\]

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