2.8.1 \(u_{xx} + y u_{yy} = 0\) with \(u(x,0)=0,u_y(x,0)=x^2\)

problem number 99

From Mathematica DSolve helps pages.

Boundary value problem for the Tricomi equation.

Solve for \(u(x,y)\) \[ u_{xx} + y u_{yy} = 0 \] With boundary conditions \begin {align*} u(x,0)&=0 \\ \frac {\partial u}{\partial y}(x,0) &= x^2 \end {align*}

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, y], {x, 2}] + y*D[u[x, y], {y, 2}] == 0; 
bc  = {u[x, 0] == 0, Derivative[0, 1][u][x, 0] == x^2}; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, bc}, u[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{u(x,y)\to -y \left (y-x^2\right )\right \}\right \}\]

Maple ✓

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

\[u \left (x , y\right ) = \left (x^{2}-y \right ) y\]

____________________________________________________________________________________