30 Cauchy Riemann PDE’s

 30.1 Cauchy Riemann PDE with Prescribe the values of \(u\) and \(v\) on the \(x\) axis
 30.2 Cauchy Riemann PDE With extra term on right side

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

problem number 167

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

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

Maple

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

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30.2 Cauchy Riemann PDE With extra term on right side

problem number 168

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

\[ \text{DSolve}\left [\left \{u^{(1,0)}(x,y)=v^{(0,1)}(x,y),u^{(0,1)}(x,y)=y-v^{(1,0)}(x,y)\right \},\{u(x,y),v(x,y)\},\{x,y\}\right ] \]

Maple

\[ \left \{ u \left ( x,y \right ) =-i{\it \_F1} \left ( y-ix \right ) +i{\it \_F2} \left ( y+ix \right ) +yx+{\it \_C1},v \left ( x,y \right ) ={\it \_F1} \left ( y-ix \right ) +{\it \_F2} \left ( y+ix \right ) +1/2\,{x}^{2} \right \} \]