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October 7th, 2019, 08:53 AM   #1
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Hard PDE

Solve $\displaystyle \frac{\partial Z }{\partial x} =2x\cdot (1-\frac{\partial Z}{\partial y})\;$ , for $\displaystyle Z(x,0)=Z(0,y)=1$ .
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October 7th, 2019, 09:58 AM   #2
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$\displaystyle \mathscr{L} (z_x) +2x \mathscr{L} (z_y )=2x \mathscr{L}(1)$.
$\displaystyle Z(x,y)=Ce^{-px^2 }+p^{-2}(1+p)=-e^{-px^2 }/p^{2}+p^{-2}(1+p)$.
$\displaystyle \mathscr{L}^{-1} (Z)=-\mathscr{L}^{-1}(p^{-2} e^{-px^2 })+\mathscr{L}^{-1} (p^{-2})+\mathscr{L}^{-1}(1/p)$.

$\displaystyle Z=-(y-x^2 )^{2} \int_{-\infty}^{x} \delta(s)ds+1+y $.
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