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October 7th, 2019, 08:53 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98  Hard PDE
Solve $\displaystyle \frac{\partial Z }{\partial x} =2x\cdot (1\frac{\partial Z}{\partial y})\;$ , for $\displaystyle Z(x,0)=Z(0,y)=1$ .

October 7th, 2019, 09:58 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 
$\displaystyle \mathscr{L} (z_x) +2x \mathscr{L} (z_y )=2x \mathscr{L}(1)$. $\displaystyle Z(x,y)=Ce^{px^2 }+p^{2}(1+p)=e^{px^2 }/p^{2}+p^{2}(1+p)$. $\displaystyle \mathscr{L}^{1} (Z)=\mathscr{L}^{1}(p^{2} e^{px^2 })+\mathscr{L}^{1} (p^{2})+\mathscr{L}^{1}(1/p)$. $\displaystyle Z=(yx^2 )^{2} \int_{\infty}^{x} \delta(s)ds+1+y $. 

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