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The database contains 327 equations (9 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-
\left(\frac{\partial w}{\partial y}\right)^2\frac{\partial^2 w}{\partial x^2}+
\left[\left(a+\frac{\partial w}{\partial x}\right)^2-
f(x)\left(\frac{\partial w}{\partial y}\right)^{-1}\right]\frac{\partial^2 w}{\partial y^2}-
f(x)\frac{\partial w}{\partial y}=0$.\hfill\break
The transformation
U=\left(a+\frac{\partial w}{\partial x}\right)\frac{\partial w}{\partial y}+\int f(x)\,dx,
\quad V=\frac{\partial w}{\partial y}, \quad T=ax+w,
leads to the equation
\frac{\partial U}{\partial T}-V\frac{\partial U}{\partial V}=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(T+\ln V)}$.
Hence, we have a first order PDE:
\left(a+\frac{\partial w}{\partial x}\right)\frac{\partial w}{\partial y}+\int f(x)\,dx=
\Phi\left(\frac{\partial w}{\partial y}e^{ax+w}\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
Statistic information
Submission date:Wed 09 Jan 2008 13:12
Edits by author:0

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