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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \frac{\partial^2 w}{\partial x\partial z}\frac{\partial^2 w}{\partial y\partial z}-
\frac{\partial^2 w}{\partial z^2}\frac{\partial^2 w}{\partial x\partial y}+
f(z)\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial y\partial z}=0$.\hfill\break
The transformation
U=\frac{\partial w}{\partial x}\exp\left(\int f(z)\,dz\right), \quad
P=x, \quad Q=\frac{\partial w}{\partial z}, \quad
R=\Psi_1\left(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y},\frac{\partial w}{\partial z}\right),
leads to the equation
\frac{\partial U}{\partial R}=0.
Its solution is $\,\displaystyle {U(P,Q,R)=\Psi(P,Q)}$.
Hence, we have a first order PDE:
\frac{\partial w}{\partial x}\exp\left(\int f(z)\,dz\right)=
\Phi\left(x,\frac{\partial w}{\partial z}\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 16:25
Edits by author:0

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