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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}+
\frac1w\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}
\left(a\frac{\partial^2 w}{\partial x^2}-\frac{\partial^2 w}{\partial x\partial y}\right)=0$.\hfill\break
The transformation
U=-\frac{aw^2}{2\left(a\frac{\partial w}{\partial x}-\frac{\partial w}{\partial y}\right)}-
\frac{w-x\frac{\partial w}{\partial x}}{\frac{\partial w}{\partial x}},
\quad V=w, \quad T=\frac{\partial w}{\partial x}-\frac1a\frac{\partial w}{\partial y},
leads to the equation
\frac{\partial U}{\partial V}-\frac1T V=0.
Its solution is $\,\displaystyle{U(V,T)=-\frac{V^2}{2T}+\Psi(T)}$.
Hence, we have a first order PDE:
\frac{\partial w}{\partial x}\left[\Phi\left(\frac{\partial w}{\partial x}-
\frac1a\frac{\partial w}{\partial y}\right)-x\right]+w=0,
where $\,\Phi\,$ 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 13:39
Edits by author:0

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