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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial x\partial y}-
\left[a\frac{\partial w}{\partial x}-g(x)\right]\frac{\partial^2 w}{\partial y^2}=0$.\hfill\break
The transformation
U=w, \quad
V=\frac{\partial w}{\partial y}, \quad
T=\sqrt{2\int g(x)\,dx-2(a+1)\left(w-y\frac{\partial w}{\partial y}\right)},
leads to the equation
\frac{\partial U}{\partial T}+T=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(V)-\frac12 T^2}$.
Hence, we have a first order PDE:
(a+1)y\frac{\partial w}{\partial y}-aw+\int g(x)\,dx=
\Phi\left(\frac{\partial w}{\partial y}\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 13:46
Edits by author:0

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