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View Equation

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

Equation data
Category:1. Ordinary Differential Equations
Subcategory:1.2. Second-Order Linear Equations
Equation(s):$\displaystyle y''_{xx} + f(x)y'_x + g(x)y = 0$.
Solution(s),
Transformation(s),
Integral(s)
:
Let $\,y_0=y_0(x)\,$ be any nontrivial particular solution
of the equation. 

$1^\circ$. The general solution can be represented as:
$$
  y = y_0\left(C_1+C_2\int\frac{e^{-F}}{y_0^2}\,dx\right),\quad \ \
  \hbox{where}\quad F = \int f(x)\,dx.
$$

$2^\circ$. The equation can be factored as:
$$
L_2L_1y=0,
$$
where
$$
L_1=\frac 1{\psi}\left(\frac{d}{dx}-\frac{y_0'}{y_0}\right),\quad \
L_2=\psi\left(\frac{d}{dx}+\frac{y_0'}{y_0}+\frac{\psi'}{\psi}+f(x)\right),
$$
and $\psi=\psi(x)$ is an arbitrary function.
Novelty:Material has been partially published elsewhere
References:1. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003. <br> 2. Berkovich, F. L., Factorization and transformations of differential equations: methods and applications [in Russian], Regulyarnaya i Haoticheskaya Dinamika, Moscow, 2002.
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
City:Moscow
Statistic information
Submission date:Mon 09 Jul 2007 14:59
Edits by author:0

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