Hard PDE

Dec 2015
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Solve \(\displaystyle \frac{\partial Z }{\partial x} =2x\cdot (1-\frac{\partial Z}{\partial y})\;\) , for \(\displaystyle Z(x,0)=Z(0,y)=1\) .
 
Dec 2015
1,084
169
Earth
\(\displaystyle \mathscr{L} (z_x) +2x \mathscr{L} (z_y )=2x \mathscr{L}(1)\).
\(\displaystyle Z(x,y)=Ce^{-px^2 }+p^{-2}(1+p)=-e^{-px^2 }/p^{2}+p^{-2}(1+p)\).
\(\displaystyle \mathscr{L}^{-1} (Z)=-\mathscr{L}^{-1}(p^{-2} e^{-px^2 })+\mathscr{L}^{-1} (p^{-2})+\mathscr{L}^{-1}(1/p)\).

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