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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}-
\frac{f'(x)}{f(x)}\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial y^2}-
The transformation
U=\left(\frac{\partial w}{\partial y}-
\int\frac{dx}{f(x)}\right)\exp\left(xw\frac{\partial w}{\partial y}\right),
\quad V=f(x)\frac{\partial w}{\partial x}+y, \quad T=xw\frac{\partial w}{\partial y},
leads to the equation
\frac{\partial U}{\partial T}-U=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(V)e^T}$.
Hence, we have a first order PDE:
\frac{\partial w}{\partial y}-\int\frac{dx}{f(x)}=
\Phi\left(f(x)\frac{\partial w}{\partial x}+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 13:16
Edits by author:0

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