18 research outputs found
Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides
After tersely reviewing the various meanings that can be given to the
property of a system of nonlinear ODEs to be solvable, we identify a special
case of the system of two first-order ODEs with homogeneous quadratic
right-hand sides which is explicitly solvable. It is identified by 2 explicit
algebraic constraints on the 6 a priori arbitrary parameters that characterize
this system. Simple extensions of this model to cases with nonhomogeneous
quadratic right-hand sides are also identified, including isochronous cases.Comment: 18 page
Explicitly Solvable Systems of First-order Ordinary Differential Equations with Polynomial Right-hand Sides, and Their Periodic Variants
In this Letter we identify special systems of (an arbitrary number) N of
first-order Ordinary Differential Equations with homogeneous polynomials of
arbitrary degree M on their right-hand sides, which feature very simple
explicit solutions; as well as variants of these systems--with right-hand sides
no more homogeneous--which feature periodic solutions. A novelty of these
findings is to consider special systems characterized by constraints involving
both their parameters and their initial data.Comment: 4 page