Equivalent Lagrangians are used to find, via transformations, solutions and conservation law of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as the Kummer equation and the combined gravity-inertial-...

This paper studies the existence of periodic solutions to the delay differential equation ẋ(t) = f(x(t− μτ(t)), ) . The analysis is based on a perturbation method previously used for retarded differential equations with constant delay. By transforming the studied equation into a perturbed non-autonomous ordinary equation and using a bifurcation result and the Poincaré procedure for this last eq...

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space ...

For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider t...

This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale Nt that solves the Engelbert–Schmidt stochastic differential equation (??). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depen...

In this paper we present some properties of the Riccatti differential polynomial associated with a homogeneous linear ordinary differential equation. We give a complete description of the differential Newton polygons of their derivatives and its evaluations.

where A is the infinitesimal generator of a compact analytic semigroup of linear operators, (T (t))t≥0, on a Banach space X; the history xt, xt(θ) = x(t+θ), belongs to an appropriate phase space D and G,F : R×D → X are continuous functions. The Massera Theorem [10] is a pioneer work in the study of the relations between the boundedness of solutions and the existence of periodic solutions. In Ma...

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or Bäcklund transformations. We describe such a chain for the sixth Painlevé equation (PVI), containing, apart from PVI itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-Bäcklund transformation, relatin...

The most up-to-date version of this collection of homework exercises can always be found at 1. Find the general solution to the following ordinary differential equation. dy dx = (y − 1)(x − 2)(y + 3) (x − 1)(y − 2)(x + 3) 2. Find the general solution to the following ordinary differential equation. x 3 e 2x 2 +3y 2 dx − y 3 e −x 2 −2y 2 dy = 0 3. Find the general solution to the following ordin...