×

The generalized Emden-Fowler equation. (English) Zbl 0952.34030

Shkil, Mykola (ed.) et al., Symmetry in nonlinear mathematical physics. Proceedings of the second international conference, Kyiv, Ukraine, July 7-13, 1997. Memorial Prof. W. Fushchych conference. Vol. 1. Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine. 155-163 (1997).
The author gives necessary and sufficient conditions when a nonlinear nonautonomous ordinary differential equation of order \(n\) with the so-called reducible linear part can be reduced to an autonomous form using the Kummer-Liouville transformation.
It is proved that the Emden-Fowler equation \[ y''+ax^{-1}y'+bx^{m-1}y^n=0, \] with \(n\not=0\) and \(n\not=1\), is invariant under the one-parameter group of local transformations with the generator \(x \partial/\partial x+(1+m)(1-n)^{-1}y \partial/\partial y\) and can be reduced to an autonomous form by means of the transformation \(y=x^{(1+m)/(1-n)}z\), \(dt=x^{-1}dx\).
Then, the class of generalized Emden-Fowler equations of the form \[ y''+a_1(x)y'+a_0(x)y+f(x)y^n=0, \] with \(n\not=0\) and \(n\not=1\), is considered. The author obtains conditions on the coefficients \(a_1(x)\), \(a_0(x)\) and \(f(x)\) when equations from this class admit Lie symmetries and can be reduced to an autonomous form. Some exact solutions of the equations admitting Lie symmetries are constructed.
For the entire collection see [Zbl 0882.00038].

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C14 Symmetries, invariants of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
PDFBibTeX XMLCite