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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-
f(x)\frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial y^2}=0$.\hfill\break
The transformation
U=\frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}\int f(x)\,dx, \quad
V=\frac{\partial w}{\partial y}, \quad
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}+\frac{\partial w}{\partial y}\int f(x)\,dx=
\Phi\left(\frac{\partial w}{\partial y}\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 13:41
Edits by author:0

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