Ordinary differential equation
[1] The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable,[2] and, less commonly, in contrast with stochastic differential equations (SDEs) where the progression is random.
For applied problems, numerical methods for ordinary differential equations can supply an approximation of the solution.
Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences.
Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modeling), chemistry (reaction rates),[7] biology (infectious diseases, genetic variation), ecology and population modeling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler.
In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs).
This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems.
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,[24] meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after.
These finite-duration solutions can't be analytical functions on the whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of Lipschitz differential equations.
As example, the equation: Admits the finite duration solution: The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century has it received special attention.
Darboux (from 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably Casorati and Cayley.
To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.
The primitive attempt in dealing with differential equations had in view a reduction to quadratures.
As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the
th degree, so it was the hope of analysts to find a general method for integrating any differential equation.
Hence, analysts began to substitute the study of functions, thus opening a new and fertile field.
Two memoirs by Fuchs[25] inspired a novel approach, subsequently elaborated by Thomé and Frobenius.
As the latter can be classified according to the properties of the fundamental curve that remains unchanged under a rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces
From 1870, Sophus Lie's work put the theory of differential equations on a better foundation.
He showed that the integration theories of the older mathematicians can, using Lie groups, be referred to a common source, and that ordinary differential equations that admit the same infinitesimal transformations present comparable integration difficulties.
Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions.
Continuous group theory, Lie algebras, and differential geometry are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform, and finally finding exact analytic solutions to DE.
Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.
There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally.
Also, uniqueness theorems like the Lipschitz one above do not apply to DAE systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone.
When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result.
there exists a unique maximum (possibly infinite) open interval such that any solution that satisfies this initial condition is a restriction of the solution that satisfies this initial condition with domain
The differential equations are in their equivalent and alternative forms that lead to the solution through integration.
are dummy variables of integration (the continuum analogues of indices in summation), and the notation
In the case of a first order ODE that is non-homogeneous we need to first find a solution to the homogeneous portion of the DE, otherwise known as the associated homogeneous equation, and then find a solution to the entire non-homogeneous equation by guessing.
