From Mathematica DSolve helps pages.
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} \end {align*}
With boundary conditions \begin {align*} u(x,0)&=x^3 \\ v(x,0)&=0 \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]; bc = {u[x, 0] == x^3, v[x, 0] == 0}; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde1, pde2, bc}, {u[x, y], v[x, y]}, {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to x^3-3 x y^2,v(x,y)\to 3 x^2 y-y^3\right \}\right \}\]
Maple ✓
restart; pde1:= diff(u(x,y),y)=diff(v(x,y),x); pde2:= diff(u(x,y),x)=-diff(v(x,y),y); bc := u(x,0)=x^3,v(x,0)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde1,pde2,bc],[u(x,y),v(x,y)])),output='realtime'));
\[ \left \{ u \left ( x,y \right ) ={x}^{3}-3\,x{y}^{2},v \left ( x,y \right ) =-3\,{x}^{2}y+{y}^{3} \right \} \]
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