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View Equation

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

Equation data
Category:1. Ordinary Differential Equations
Subcategory:1.7. Nonlinear Systems of Three and More Equations
Equation(s):\noindent
$\displaystyle y'_i=f_i(t,y_1,\dots,y_n),\qquad i=1,\dots,n$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
1. The formal exact general solution in the operator form:\hfill\break
$\displaystyle y_i(t)=\left[\exp \left\{
(t-a)\,\left[\frac{\partial}{\partial s}+\sum_{i=1}^n\,f_i(s,\vec{c})\frac{\partial}{\partial
c_i}\right]\right\} \,c_i\,\right]|_{s=a},\qquad (i=1,\dots,n)$,\hfill\break
where $\vec{c}=(c_1,\dots,c_n)$ is the set of arbitrary constants.
\medskip

\noindent
2. If expand the operator exponents into Taylor series and execute operations, one can obtain the conventional formal power series to solutions of given system of differential equations.
Remarks:\noindent
The consistent system of higher-order ODEs can be always redused to the first-order system by introducing some auxiliary functions for second and higher-order derivatives of unknown functions.
Novelty:Material has been fully published elsewhere
References:Yu.N. Kosovtsov, The Chronological Operator Algebra and Formal Solutions of Differential Equations, 2004, http://arxiv.org/abs/math-ph/0409035. See also Maple implementations of the operator method at http://www.maplesoft.com/applications/app_center_advanced_search.aspx?ABA=325
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Sun 10 Dec 2006 12:32
Edits by author:3
Last edit by author:Fri 15 Dec 2006 09:34

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