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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:4.7. Systems of Two Equations
Equation(s):$\displaystyle \begin{array}[c]{ll}
F_1\left(au^k+bw,\cfrac{u_x}{u},\cfrac{u_{xx}}{u},\dots,\cfrac{u_x^{(m)}}{u},
\cfrac{w_t}{u^k},\cfrac{w_{tt}}{u^k},\dots,\cfrac{w_t^{(n)}}{u^k}\right)&=0,\cr
F_2\left(au^k+bw,\cfrac{u_x}{u},\cfrac{u_{xx}}{u},\dots,\cfrac{u_x^{(m)}}{u},
\cfrac{w_t}{u^k},\cfrac{w_{tt}}{u^k},\dots,\cfrac{w_t^{(n)}}{u^k}\right)&=0.
\end{array}$
Solution(s),
Transformation(s),
Integral(s)
:
Solution:\hfill\break
$\displaystyle u=e^{\lambda t}\theta(x),\quad \ w=\varphi(x)-ae^{k\lambda
t}[\theta(x)]^k,$\hfill\break
where $\theta(x)$ and $\varphi(x)$ are described by the system of 
ordinary differential equations:
\hfill\break
$\displaystyle \begin{array}[c]{ll}
F_1\left(b\varphi,\cfrac{\theta_x}{\theta},\cfrac{\theta_{xx}}{\theta},
\dots,\cfrac{\theta_x^{(m)}}{\theta},
-ak\lambda,-a(k\lambda)^2,\dots,-a(k\lambda)^m\right)&=0,\cr
F_2\left(b\varphi,\cfrac{\theta_x}{\theta},\cfrac{\theta_{xx}}{\theta},
\dots,\cfrac{\theta_x^{(m)}}{\theta},
-ak\lambda,-a(k\lambda)^2,\dots,-a(k\lambda)^m\right)&=0.\end{array}$\hfill\break
Remarks:Here $F_1$ and $F_2$ are arbitrary functions of their respective arguments.
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Vyazmina
First name:Elena
Country:Russia
City:Moscow
Affiliation:Institute for Problems in Mechanics
Statistic information
Submission date:Sun 23 Sep 2007 10:51
Edits by author:0

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