2.8.2 \(u_{xx} + x u_{yy} = 0\)

problem number 100

Added June 20, 2019

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

Solve for \(u(x,y)\)

\[ u_{xx} + x u_{yy} = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

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