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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):\begin{multline*} \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}+
\left\{[f(w)-g(w)]\frac{\partial w}{\partial x}+[f(w)+g(w)]\frac{\partial w}{\partial y}\right\}
\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial x\partial y}+\\
+\left\{[g(w)-f(w)]\frac{\partial w}{\partial x}-f(w)\frac{\partial w}{\partial y}\right\}
\frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial x^2}-
g(w)\left(\frac{\partial w}{\partial x}\right)^2\frac{\partial^2 w}{\partial y^2}=0.
The transformation
U=\frac{\partial w}{\partial x}\exp\left(\int g(w)\,dw\right), \quad
V=\left(\frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}\right)\exp\left(\int f(w)\,dw\right),
T=\Psi_1\left(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}\right),
leads to the equation
\frac{\partial U}{\partial T}=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(V)}$.
Hence, we have a first order PDE:
\frac{\partial w}{\partial x}\exp\left(\int g(w)\,dw\right)=
\Phi\left(\left(\frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}\right)\exp\left(\int f(w)\,dw\right)\right),
where $\Phi$ is an arbitrary function.
Remarks:This result was obtained in co-authorship with L.V. Linchuk,
using generalized group analysis. The main idea of the method 
is outlined at EqWorld (\slash 
ru/solutions/interesting.htm, in Russian).
Novelty:New equation(s) & solution(s) & transformation(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Statistic information
Submission date:Wed 09 Jan 2008 13:40
Edits by author:0

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