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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}-

\textit{Monge--Amp\`ere equation}
The transformation
U=\Psi_1\left(\frac{\partial w}{\partial x}+\frac{a}{y+b},\,\,\, x\frac{\partial w}{\partial x}+
(y+b)\frac{\partial w}{\partial y}-u+\frac{ax}{y+b}\right),
\quad V=y, \quad T=\frac{\partial w}{\partial x},
$\Psi_1(z_1,z_2)\,$ is arbitrary function,
leads to the equation
(V+b)^2\frac{\partial U}{\partial V}+a\frac{\partial U}{\partial T}=0,
Its solution is $\,\displaystyle{U=\Phi_1\left(T+\frac{a}{V+b}\right)}$.
Hence, we have a first order PDE:
\Psi\left(\frac{\partial w}{\partial x}+\frac{a}{y+b},\,\,\, x\frac{\partial w}{\partial x}+
(y+b)\frac{\partial w}{\partial y}-u+\frac{ax}{y+b}\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:Tue 04 Sep 2007 18:01
Edits by author:0

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