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View Equation

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

Equation data
Category:1. Ordinary Differential Equations
Subcategory:1.5. Higher-Order Nonlinear Equations
Equation(s):\noindent
$\displaystyle y^{(n)}_x=f(x,y,y'_x,\dots,y^{(n-1)}_x)$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
1. Formal exact general solution in operator form:\hfill\break
$\displaystyle y(x)=\left[\exp \left\{ (x-a) \,\left[f(s,\vec{c})\frac{\partial}{\partial
c_{n-1}}+\dots+c_{i}\frac{\partial}{\partial
c_{i-1}}+\dots+c_1\frac{\partial}{\partial
c_0}+\frac{\partial}{\partial s}\right]\right\} \,c_0\,\right]\,|_{s=a}$,\hfill\break
where $\vec{c}=(c_0,...,c_{n-1})$ is the set of arbitrary constants, with $c_k=y^{(k)}_x|_{x=a}$.
\medskip

\noindent
2. On expanding the operator exponent into Taylor series and executing operations, one arrives at the conventional formal series solution of the given ODE in powers of $(x-a)$.
Remarks:Notation: $\displaystyle y^{(n)}_x=\frac{d^ny}{dx^n}$.
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:Fri 08 Dec 2006 10:07
Edits by author:1
Last edit by author:Fri 15 Dec 2006 09:43

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