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View Equation

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

Equation data
Category:1. Ordinary Differential Equations
Subcategory:1.3. Second-Order Nonlinear Equations
Equation(s):\noindent
$\displaystyle [x+(2y-xy'_x)^3(y'_x)^2]y''_{xx}=y'_x$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The first integral is
$$
2(y'_x)^3+\frac{3}{(2y-xy'_x)^2}=C,
\eqno(*)
$$
where $C$ is an arbitrary constant. By the Legendre transformation $\,x=p'_q$, $\,y=qp'_q-p$, $\,y'_x=q$, where $\,p=p(q)$, 
equation (*) can be reduced to the first-order linear equation
$$
p'_q=\frac 2q\, p\pm\frac{\sqrt{6}}{q\sqrt{-q^3-C}}.
$$
Novelty:Material has been fully published elsewhere
References:Linchuk L.V., Zaitsev V.F On searching technologies of symmetries of ordinary differential equations // Izvestiya RGPU \No 5 (13), 2005 -- pp. 38--49.
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Country:Russia
City:St. Petersburg
Affiliation:PGPU
Statistic information
Submission date:Mon 11 Dec 2006 08:53
Edits by author:0

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