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In mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. There are many general forms an ODE can take, and these are classified in practice (see below). The derivatives are ordinary because partial derivatives only apply to functions of many independent variables (see Partial differential equation).
The subject of ODEs is a sophisticated one (more so with PDEs), primarily due to the various forms the ODE can take and how they can be integrated. Linear differential equations are ones with solutions that can be added and multiplied by coefficients, and the theory of linear differential equations is well-defined and understood, and exact closed form solutions can be obtained. By contrast, ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary functions in closed form — rather the exact (or "analytic") solutions are in series or integral form. As an alternative to the exact analytic solution, graphical and numerical methods (by hand or on computer) may be used to generate approximate solutions. The properties of such approximate solutions may yield very useful information, which often suffices in the absence of the exact analytic solution.
Ordinary differential equations (ODEs) arise in many different contexts throughout mathematics and science (social and natural) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, describing dynamical phenomena, evolution and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations.
Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), geology (weather modelling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).
A simple example is Newton's second law of motion — the relationship between the displacement x and the time t of the object under the force F which leads to the differential equation
for the motion of a particle of constant mass m. In general, F depends on the position x(t) of the particle at time t, and so the unknown function x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).
In what follows, let y be a dependent variable and x an independent variable, so that y = y(x) is an unknown function in x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context the Leibniz's notation (dy/dx,d2y/dx2,...dny/dxn) is useful for differentials and when integration is to be done, while Newton's and Lagrange's notation (y′,y′′, ... y(n)) is useful for representing derivatives of any order compactly.
Let F be a given function of x, y, and derivatives of y. Then an equation of the form
There are further classifications:
A differential equation not depending on x is called autonomous.
A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y:
where ai(x) and r(x) continuous functions in x. Non-linear equations cannot be written in this form. The function r(x) is called the source term, leading to two further important classifications:
Homogeneous: If r(x) = 0, and consequently one "automatic" solution is the trivial solution, y = 0. The solution of a linear homogeneous equation is a complementary function, denoted here by yc.
Nonhomogeneous (or inhomogeneous): If r(x) ≠ 0. The additional solution to the complementary function is the particular integral, denoted here by yp.
The general solution to a linear equation can be written as y = yc + yp.
A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(x) = [y1(x), y2(x),..., ym(x)], and F is a vector valued function of y and its derivatives, then
is an explicit system of ordinary differential equations of order or dimension m. In column vector form:
These are not necessarily linear. The implicit analogue is:
where 0 = (0, 0,... 0) is the zero vector. In matrix form
Given a differential equation
a function u: I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and
Given two solutions u: J ⊂ R → R and v: I ⊂ R → R, u is called an extension of v if I ⊂ J and
A solution which has no extension is called a maximal solution. A solution defined on all of R is called a global solution.
A general solution of an n-th order equation is a solution containing n arbitrary independent constants of integration. A particular solution is derived from the general solution by setting the constants to particular values, often chosen to fulfill set 'initial conditions or boundary conditions'. A singular solution is a solution which cannot be obtained by assigning definite values to the arbitrary constants in the general 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 did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was 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 nth degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.
Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations.
From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.
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. The theory has applications to both ordinary and partial differential equations.
A general approach to solve DE's uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the DE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
Sturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linear operators defined in terms of second-order homogeneous linear equations, and is useful in the analysis of certain partial differential equations.
There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are
|Peano existence theorem||F continuous||local existence only|
|Picard–Lindelöf theorem||F Lipschitz continuous||local existence and uniqueness|
which are both local results.
The theorem can be stated simply as follows. For the equation and initial value problem:
if F and ∂F/∂y are continuous in a closed rectangle
for some h ∈ ℝ where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F to be linear, this applies to non-linear equations which take the form F(x, y), and it can also be applied to systems of equations.
For each initial condition (x0, y0) there exists a unique maximum (possibly infinite) open interval
such that any solution which satisfies this initial condition is a restriction of the solution which satisfies this initial condition with domain Imax.
In the case that , there are exactly two possibilities
where Ω is the open set in which F is defined, and is its boundary.
Note that the maximum domain of the solution
This means that F(x, y) = y2, which is C1 and therefore Lipschitz continuous for all y, satisfying the Picard–Lindelöf theorem.
Even in such a simple setting, the maximum domain of solution cannot be all ℝ, since the solution is
which has maximum domain:
This shows clearly that the maximum interval may depend on the initial conditions. The domain of y could be taken as being , but this would lead to a domain that is not an interval, so that the side opposite to the initial condition would be disconnected from the initial condition, and therefore not uniquely determined by it.
The maximum domain is not ℝ because
which is one of the two possible cases according to the above theorem.
Differential equations can usually be solved more easily if the order of the equation can be reduced.
Any differential equation of order n,
can be written as a system of n first-order differential equations by defining a new family of unknown functions
for i = 1, 2,... n. The n-dimensional system of first-order coupled differential equations is then
more compactly in vector notation:
Some differential equations have solutions which can be written in an exact and closed form. Several important classes are given here.
In the table below, P(x), Q(x), P(y), Q(y), and M(x,y), N(x,y) are any integrable functions of x, y, and b and c are real given constants, and C1, C2,... are arbitrary constants (complex in general). The differential equations are in their equivalent and alternative forms which lead to the solution through integration.
In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in summation), and the notation ∫xF(λ)dλ just means to integrate F(λ) with respect to λ, then after the integration substitute λ = x, without adding constants (explicitly stated).
|Differential equation||Solution method||General solution|
|First order, separable in x and y (general case, see below for special cases)||Separation of variables (divide by P2Q1).|
|First order, separable in x||Direct integration.|
|First order, autonomous, separable in y||Separation of variables (divide by F).|
|First order, separable in x and y||Integrate throughout.|
|General first order equations|
|First order, homogeneous||Set y = ux, then solve by separation of variables in u and x.|
|First order, separable||Separation of variables (divide by xy).|
If N = M, the solution is xy = C.
|Exact differential, first order|
where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation.
|Inexact differential, first order|
|Integration factor μ(x, y) satisfying ||If μ(x, y) can be found: |
|General second order equations|
|Second order, autonomous||Multiply equation by 2dy/dx, substitute , then integrate twice.|
|Linear equations (up to nth order)|
|First order, linear, inhomogeneous, function coefficients||Integrating factor: .|
|Second order, linear, inhomogeneous, constant coefficients||Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions .|
If b2 > 4c, then:
If b2 = 4c, then:
If b2 < 4c, then:
|nth order, linear, inhomogeneous, constant coefficients||Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions .|
for αj all different,
for each root αj repeated kj times,
for some αj complex, then setting α = χj + iγj, and using Euler's formula, allows some terms in the previous results to be written in the form
where ϕj is an arbitrary constant (phase shift).
|Wikibooks has a book on the topic of: Calculus/Ordinary differential equations|