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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
$\displaystyle \left[(a_2b_1-a_1b_2)w-a_1b_3+b_1a_3\right]\frac{\partial^2 w}{\partial t\partial x} &=
\left[(a_2b_1-a_1b_2)\frac{\partial w}{\partial t}+
a_2b_3-a_3b_2\right]\,\frac{\partial w}{\partial x}\notag \\\notag \\
&-(b_1\frac{\partial w}{\partial t}+b_2w+b_3)^2\,F \left(\frac{a_1\frac{\partial w}{\partial t}+a_2w+a_3}{b_1\frac{\partial w}{\partial t}+b_2w+b_3}\right)$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
General solution:
$$
w =\exp\left[\int\frac{-a_2+b_2Y(t,x)}{a_1-b_1Y(t,x)}\,dt\right]
\left\{\int\frac{-a_3+b_3Y(t,x)}{a_1-b_1Y(t,x)}&\exp\left[-\int\frac{-a_2+b_2Y(t,x)}{a_1-b_1Y(t,x)}\,dt\right]\,dt+\Phi(x)\right\},
$$
where the function $Y(t,x)$ is determined by the trancendental equation
$$
\int^{Y}_c\frac{dz}{F(z)}=x+\Psi(t),
$$ 
and $\Phi(x)$ and $\Psi(t)$ are arbitrary functions,  $c$ is any constant.
Remarks:Here, $F(z)\neq 0$ is an arbitrary function.
Novelty:New equation(s) & solution(s) / integral(s)
References:Particular cases of this PDEs family see at http://arxiv.org/abs/math-ph/0609003 (sec.3).
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Fri 08 Dec 2006 09:02
Edits by author:2
Last edit by author:Wed 20 Dec 2006 11:02

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