Solve for \(u(x,y),v(x,y\) \begin {align*} \frac {\partial u}{\partial x} &= \frac {\partial v}{\partial y}\\ \frac {\partial u}{\partial y} &= -\frac {\partial v}{\partial x} + y \end {align*}
Mathematica ✗
ClearAll["Global`*"]; ClearAll[u, v, x, y]; pde1 = D[u[x, y], x] == D[v[x, y], y]; pde2 = D[u[x, y], y] == -D[v[x, y], x] + y; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde1, pde2}, {u[x, y], v[x, y]}, {x, y}], 60*10]];
Failed
Maple ✓
restart; pde1:= diff(u(x,y),y)=diff(v(x,y),x); pde2:= diff(u(x,y),x)=-diff(v(x,y),y)+y; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde1,pde2],[u(x,y),v(x,y)])),output='realtime'));
\[\left \{u \left (x , y\right ) = x y +c_{1}-i \textit {\_F1} \left (-i x +y \right )+i \textit {\_F2} \left (i x +y \right ), v \left (x , y\right ) = \frac {x^{2}}{2}+\textit {\_F1} \left (-i x +y \right )+\textit {\_F2} \left (i x +y \right )\right \}\]