Solve-for-X-x-y-z-Y-x-y-z-Z-x-y-z-Z-y-Y-z-1-x-2-Z-x-X-z-y-2-2-Y-x-X-y-z-2x-y-where-X-x-y-0-0-Y-x-y-0-0-Z-x-y-0-0-

Question Number 110262 by Ar Brandon last updated on 28/Aug/20

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right) \\ $$$$\begin{cases}{\frac{\partial\mathrm{Z}}{\partial\mathrm{y}}−\frac{\partial\mathrm{Y}}{\partial\mathrm{z}}=\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\\{\frac{\partial\mathrm{Z}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{z}}=−\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\\{\frac{\partial\mathrm{Y}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{y}}=\mathrm{z}\left(\mathrm{2x}−\mathrm{y}\right)}\end{cases}\:\mathrm{where}\:\begin{cases}{\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\end{cases} \\ $$

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