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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \left(y\frac{\partial w}{\partial y}+aw\right)\frac{\partial^2 w}{\partial x\partial y}
-y\left[\frac{\partial w}{\partial x}+f(x)w\right]\frac{\partial^2 w}{\partial y^2}=0$,\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
The equation is represented in equivalent form
$$
\frac{\partial U}{\partial T}=0,
\eqno(1)
$$
where
$$
U=\frac{\partial w}{\partial y}, \quad V=y^a w\exp\int f(x)\,dx, \quad T=x,
$$
The solution of Eq.(1) is $\,U=\Phi(V)$. Hence, we have a first order PDE:
$$
\frac{\partial w}{\partial y}=\Phi\left(y^a w\exp\int f(x)\,dx\right),
$$
where $\,\Phi(z)\,$ is arbitrary function.
Remarks:This result was received by generalized group analysis. The main idea of the method
is announced in http://eqworld.ipmnet.ru/ru/solutions/interesting.htm
Novelty:New solution(s) & transformation(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Mon 02 Jul 2007 08:33
Edits by author:0

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