2.12.1 Cauchy Riemann PDE with Prescribe the values of \(u\) and \(v\) on the \(x\) axis

problem number 106

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*}


\[\left \{\left \{u(x,y)\to x^3-3 x y^2,v(x,y)\to 3 x^2 y-y^3\right \}\right \}\]


\[\{u \left (x , y\right ) = x^{3}-3 x y^{2}, v \left (x , y\right ) = -3 x^{2} y +y^{3}\}\]