2.16.3 \( u_{xx} + y u_{yy} = 0\)

problem number 142

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Laplace like PDE with polynomial solution. 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, eval(diff(u(x,y),y),y=0)=x^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, bc], u(x, y))),output='realtime'));
 

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

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