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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-
(y+a)^{-2}\left(\frac{\partial w}{\partial x}+b\right)^2=0$.\hfill\break
The transformation
U=\Psi_1\left(\frac{\frac{\partial w}{\partial x}+1}{y+1},\,\,\,
\frac{(y+1)\frac{\partial w}{\partial y}+x\left(\frac{\partial w}{\partial x}+1\right)}{(y+1)}\right),
\quad V=\frac{y+1}{\frac{\partial w}{\partial x}+1}, \quad T=w,
$\Psi_1(z_1,z_2)\,$ is arbitrary function,
leads to the equation
\frac{\partial U}{\partial T}=0.
Its solution is $\,\displaystyle{U(V,T)=\Phi(V)}$.
Hence, we have a first order PDE:
\Psi_1\left(\frac{1}{y+1}\left(\frac{\partial w}{\partial x}+1\right),\,\,\,
\frac{\partial w}{\partial y}+\frac{x}{y+1}\left(\frac{\partial w}{\partial x}+1\right)\right)=0,
where $\,\Psi(z_1,z_2)\,$ is arbitrary function.
Remarks:This result was received by generalized group analysis. The main idea of the method
is announced in
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Statistic information
Submission date:Wed 05 Sep 2007 13:13
Edits by author:0

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