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View Equation

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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-
a\left(\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}+
aw\frac{\partial w}{\partial x}+
\frac{\partial w}{\partial y}\right)\frac{\partial^2 w}{\partial x^2}-
a^2\left(\frac{\partial w}{\partial x}\right)^2=0$.\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
The transformation
$$
U=\Psi_1\left(e^{-ay}\frac{\partial w}{\partial x},\,\,\,
\left(\frac{\partial w}{\partial y}+aw\right)\exp\left(\frac{\partial w}{\partial x}\right)\right),
\quad V=\frac{\partial w}{\partial y}\exp\left(\frac{\partial w}{\partial x}\right),
\quad T=w\exp\left(\frac{\partial w}{\partial x}\right),
$$
$\Psi_1(z_1,z_2)\,$ is arbitrary function,
leads to the equation
$$
a\frac{\partial U}{\partial T}-\frac{\partial U}{\partial V}=0.
$$
Its solution is $\,\displaystyle{U(V,T)=\Phi(aV+T)}$.
Hence, we have a first order PDE:
$$
\Psi_1\left(e^{-ay}\frac{\partial w}{\partial x},\,\,\,
\left(\frac{\partial w}{\partial y}+aw\right)\exp\left(\frac{\partial w}{\partial x}\right)\right)=0,
$$
where $\,\Psi(z_1,z_2)\,$ is arbitrary function.
Remarks:This result was received by generalized group analysis. The main idea of the method
is announced in http://eqworld.ipmnet.ru/ru/solutions/interesting.htm
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Wed 05 Sep 2007 13:30
Edits by author:0

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