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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):\noindent
$\displaystyle \frac{\partial^2 w}{\partial t\partial x}-\frac{a}{w}\left(\frac{\partial
w}{\partial x}\right)^{\!2}-\left(\frac{1}{w}\frac{\partial w}{\partial
t}+b+\frac{c}{w}\right)\frac{\partial w}{\partial x}
-\frac{c}{2aw}\frac{\partial w}{\partial
t}-kw-\frac{bc}{2a}-\frac{c^2}{4aw} = 0$.
%where $a\neq0$, $b$, $c$, and $k$ are constants; $b^2-4ka\neq0$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The general solution:\hfill\break
$\displaystyle w(t,x) =\notag\\\notag\\&
-\frac{c}{2a}\left\{\int\exp\left[\frac{1}{2a}\int\frac{\exp(t\sqrt{b^2-4ak})G(x)(b+\sqrt{b^2-4ak})-\sqrt{b^2-4ak}+b}{1+\exp(t\sqrt{b^2-4ak})G(x)}dx\right]dx
+F(t)\right\} \times \notag\\\notag\\&
\times
\exp\left[-\frac{1}{2a}\int\frac{\exp(t\sqrt{b^2-4ak})G(x)(b+\sqrt{b^2-4ak})-\sqrt{b^2-4ak}+b}{1+\exp(t\sqrt{b^2-4ak})G(x)}dx\right]
$,\hfill\break
where $F(t)$ and $G(x)$ are arbitrary functions, and $a\neq0$ and $b^2-4ka\neq0$.
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 10:54
Edits by author:0

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