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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):\noindent
$\displaystyle 
\frac{\partial^2 w}{\partial x\partial t}-\left(\frac{1}{w}\frac{\partial w}{\partial t}+b\right)
\frac{\partial w}{\partial x}-\frac{c}{w}\frac{\partial w}{\partial t}-
aw^2\left(c+kw+\frac{\partial w}{\partial x}\right)^{\!-1}-sw-b c = 0$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The general solution:
$$
w(x,t) = e^{-kx}\exp\left[\int Z(x,t)\,dx\right]\left\{-c\int e^{kx}\exp\left[-\int
Z(x,t)\,dx\right]\,dx+F(t)\right\},
$$
where $F(t)$ and $G(x)$ are arbitrary functions, and the $Z=Z(x,t)$ is any root of the transcendental equation
$$
\int^Z_0\frac{\xi\,d\xi}{(s- bk)\xi+b\xi^2+a}=t+G(x).
$$
Novelty:Material has been fully published elsewhere
References:Yu.N. Kosovtsov. The general solutions of some nonlinear second and third order PDEs with constant and nonconstant parameters, 2006, http://arxiv.org/abs/math-ph/0609003
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Mon 11 Dec 2006 11:44
Edits by author:0

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