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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
$\displaystyle [(2x-9y)y'_x+3y]y''_{xx}=2(1-3y'_x)(y'_x)^2$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
$1^\circ$. The equation has the factorization
$$
\left\{\begin{array}{l}
t=y'_x,\\
\displaystyle u=\frac{x(y'_x)^2+(x-3y)y'_x+y}{1-3y'_x},\\
\displaystyle u'_t=\frac{3(t-1)u}{t(3t-1)}.
\end{array}\right.
$$
Solving the last equation of the system we received
$$
x(y'_x)^2+(x-3y)y'_x+y=\frac{C(y'_x)^2}{(3y'_x-1)^2},
\eqno(*)
$$
where $C$ is an arbitrary constant.
The Legendre transformation \, \mbox{$x=p'_q$}, \, \mbox{$y=qp'_q-p$}, \, \mbox{$y'_x=q$}, where $\,p=p(q)$, leads to
the first-order linear equation. 
\medskip

\noindent
$2^\circ$. 
Particular solution [for $\,C=0\,$ in equation (*)]:
$$
y=\frac{x(2x^2+3C_1)+2(x^2+C_1)^{3/2}}{9C_1},
$$
where $C_1$ is an arbitrary constant.
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:RGPU
Statistic information
Submission date:Mon 11 Dec 2006 09:01
Edits by author:0

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