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

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial x\partial y}-
\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial y^2}+
f(y)\frac{\partial w}{\partial x}g\left(\frac{\partial w}{\partial y}\right)=0=0$.\hfill\break
The transformation
U=\frac{\partial w}{\partial y}\exp\left(\frac{\partial w}{\partial x}\right), \quad
T=-w^{-1}\frac{\partial w}{\partial y}\exp\left(\frac{\partial w}{\partial x}\right)+
\exp\left[\int\frac{d\left(\frac{\partial w}{\partial y}\right)}{g\left(\frac{\partial w}{\partial y}\right)}
-\int f(y)\,dy\right]
leads to the equation
\frac{\partial U}{\partial T}+V=0.
Its solution is $\,\displaystyle{U(V,T)=-VT+\Psi(V)}$.
Hence, we have a first order PDE:
w\exp\left[\int\frac{d\left(\frac{\partial w}{\partial y}\right)}{g\left(\frac{\partial w}{\partial y}\right)}
-\int f(y)\,dy\right]=\Phi(w),
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 13:50
Edits by author:0

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