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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial y\partial z}+
\frac{\partial w}{\partial z}\right)\frac{\partial^2 w}{\partial x^2}+
\left[f(z)-\frac{\partial^2 w}{\partial x\partial z}\right]
\left(\frac{\partial^2 w}{\partial x\partial y}+\frac{\partial w}{\partial x}\right)=0$.\hfill\break
The transformation
U=\frac{\partial w}{\partial y}+w, \quad
P=\exp\left(\int f(z)\,dz\right), \quad Q=\frac{\partial w}{\partial x},
leads to the equation
P\frac{\partial U}{\partial P}+\frac{\partial U}{\partial Q}=0.
Its solution is $\,\displaystyle {U(P,Q)=\Psi(Q-\ln P)}$.
Hence, we have a first order PDE:
\frac{\partial w}{\partial y}+w=
\Phi\left(\frac{\partial w}{\partial x}-\int f(z)\,dz\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 16:29
Edits by author:0

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