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

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.6. Higher-Order Equations
Equation(s):\noindent
$\displaystyle \frac{\partial w}{\partial t}=
a\frac{\partial^n w}{\partial x^n}+(b\ln w+c)\frac{\partial w}{\partial x}$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
Solution:\hfill\break
$\displaystyle w(x,t)=e^{\lambda t}u(z),\quad \ z=x+\tfrac12b\lambda t^2+kt$,\hfill\break
where $k$, $\lambda$ are arbitrary constants and the function $u(z)$ is determined by the autonomous ordinary differential equation\hfill\break
$au^{(n)}_z+(b\ln u+c-k)u'_z-\lambda u=0$.
Remarks:\noindent
1. Notation: $w^{(n)}_x=\frac{\partial^nw}{\partial x^n}$.

\noindent
2. For an other solution see Polyanin A. D., Zaitsev V. F., 
{\it Handbook of Nonlinear Partial Differential Equations}, Chapman \& Hall/CRC Press, 
Boca Raton, 2004 (p.~636).
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Sun 10 Dec 2006 13:54
Edits by author:0

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