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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle (ax-ay-b)^2\left[\frac{\partial^2 w}{\partial x^2}
\frac{\partial^2 w}{\partial y^2}-\left(\frac{\partial^2 w}{\partial x\partial y}\right)^2\right]+
c\left(\frac{\partial^2 w}{\partial x^2}+
2\frac{\partial^2 w}{\partial x\partial y}+
\frac{\partial^2 w}{\partial y^2}\right)+\left(a\frac{\partial w}{\partial x}+
a\frac{\partial w}{\partial y}+d\right)^2=0$.\hfill\break
The transformation
U=a\left[(ax-ay-b)\frac{\partial w}{\partial y}+aw+dx\right]-c\ln(ax-ay-b),
V=(b-ax+ay)\frac{\partial w}{\partial x}+aw+x+dy-\frac{c}{a}\ln(ax-ay-b), \quad T=x,
leads to the equation
\frac{\partial U}{\partial T}+\frac{\partial U}{\partial V}=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(T-V)}$.
Hence, we have a first order PDE:
a\left[(ax-ay-b)\frac{\partial w}{\partial y}+aw+dx\right]-c\ln(ax-ay-b)=\\
=\Phi\left(a(ax-ay-b)\frac{\partial w}{\partial x}+c\ln(ax-ay-b)-a^2 w-ady\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 13:33
Edits by author:0

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