MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

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
$\displaystyle \frac{\partial^2 w}{\partial t\partial x}=
\frac{a}{w}\left(\frac{\partial w}{\partial x}\right)^{\!2}+\frac 1w\frac{\partial w}{\partial t}\frac{\partial w}{\partial x}
+\left(b+\frac{c}{w}\right)\frac{\partial w}{\partial x}+
\frac{c}{2aw}\frac{\partial w}{\partial t}+\frac{(bw+c)^2}{4aw}$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The general solution for $a\not=0$:\hfill\break
$\displaystyle w(t,x) =
\left\{-\frac{c}{2a}\int\exp\left[\frac{1}{2a}\int\frac{2\,dx}{t+G(x)}+bx\right]dx+F(t)\right\}
\exp\left[-\frac{1}{2a}\int\frac{2\,dx}{t+G(x)}+bx\right]
$,\hfill\break
where $F(t)$ and $G(x)$ are arbitrary functions.
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 10:33
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin