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The database contains 327 equations (9 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.5. Third-Order Equations
Equation(s):$\displaystyle \left[\frac{\partial^2 w}{\partial x\partial y}+
g(y)\frac{\partial w}{\partial x}\right]\frac{\partial^3 w}{\partial x\partial y^2}-
\left[\frac{\partial^2 w}{\partial x^2}+
f(x)\frac{\partial w}{\partial x}\right]\frac{\partial^3 w}{\partial y^3}=0$.\hfill\break
The transformation
U=\frac{\partial w}{\partial x}\exp\left(\int f(x)\,dx+\int g(y)\,dy\right), \quad
V=\frac{\partial^2 w}{\partial y^2}, \quad
T=\Psi_1\left(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}\right),
leads to the equation
\frac{\partial U}{\partial T}=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi(V)}$.
Hence, we have a second order PDE:
\frac{\partial w}{\partial x}\exp\left(\int f(x)\,dx+\int g(y)\,dy\right)=
\Phi\left(\frac{\partial^2 w}{\partial y^2}\right),
where $\Phi$ is 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 16:38
Edits by author:0

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