Related papers: Autonomous first order differential equations
We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
We present a general procedure to solve the equations of motion for cosmological models driven by real scalar fields with first-order differential equations. The method seems to have great power, since it works for closed, flat or open…
For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the…
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;…
Let $\mathbb{F}_q$ be a finite field with $q=p^t$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ over $\mathbb{F}_q$. A classic well-konwn result from Weil yields a…
We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class there is a first-order sentence which characterizes this field within the class up to…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
It is shown that a given non-autonomous system of two first-order ordinary differential equations can be expressed in Hamiltonian form. The derivation presented here allow us to obtain previously known results such as the infinite number of…
It is well known that every modular form~$f$ on a discrete subgroup $\Gamma\leqslant \textrm{SL}(2, \mathbb R)$ satisfies a third-order nonlinear ODE that expresses algebraic dependence of the functions~$f$, $f'$, $f''$ and~$f'''$. These…
We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=\lambda\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter.…
Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and…
A new approach to the problem of group classification is applied to the class of first-order non-linear equations of the form $u_a u_a=F(t,u,u_t)$. It allowed complete solution of the group classification problem for a class of equations…
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
The method of constructing approximate solutions of the first boundary value problem for linear differential equations based on incomplete (even and odd) trigonometric splines is considered. The theoretical positions are illustrated by…
We show that with a few modifications the Adomian's method for solving second order differential equations can be used to obtain the known results of the special functions of mathematical physics. The modifications are necessary in order to…
The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed. The representation for the corresponding matrix performing this…
This paper is the first in a sequence on the structure of sets of solutions to systems of equations over a free associative algebra. We start by constructing a Makanin-Razborov diagram that encodes all the homogeneous solutions to a…
We derive an exact solution for a simple non-autonomous delay differential equation (DDE) over the entire real-time axis, representing it as a sum of Gaussian-shaped dynamics with distinct peak positions. This marks the first explicit…
We study the most general class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the complex H\"older space $C^{n+r,\alpha}$, with $0\leq n\in\mathbb{Z}$ and…