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

problem number 103

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'));
 

\[u \left ( x,y \right ) =\sqrt {x}{x}^{-\beta /2} \left ( \BesselJ \left ( \beta /2-1/2,\sqrt {-{\it \_c}_{{1}}}x \right ) {\it \_C1}+\BesselY \left ( \beta /2-1/2,\sqrt {-{\it \_c}_{{1}}}x \right ) {\it \_C2} \right ) \left ( {\it \_C3}\,\sin \left ( \sqrt {{\it \_c}_{{1}}}y \right ) +{\it \_C4}\,\cos \left ( \sqrt {{\it \_c}_{{1}}}y \right ) \right ) \]

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