where , ..., and are arbitrary
differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function y of the variable x.
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most
special functions that are encountered in
applied mathematics are solutions of linear differential equations (see
Holonomic function). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example
trajectory of a
projectile launched from a
cannon follows a curve determined by an ordinary differential equation that is derived from Newton's second law.
Ordinary differential equations (ODEs) arise in many contexts of mathematics and
natural sciences. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.
A simple example is
Newton's second law of motion — the relationship between the displacement x and the time t of an object under the force F, is given by the differential equation
which constrains the
motion of a particle of constant mass m. In general, F is a function of the position x(t) of the particle at time t. The unknown function x(t) appears on both sides of the differential equation, and is indicated in the notation F(x(t)).
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 ordern and dimensionm. 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
The behavior of a system of ODEs can be visualized through the use of a
Given a differential equation
a function u: I ⊂ R → R, where I is an interval, is called a 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 that 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 nth-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 that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.
In the context of linear ODE, the terminology particular solution can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the homogeneous solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the
guessing method section in this article, and is frequently used when discussing the
method of undetermined coefficients and
variation of parameters.
Solutions of finite duration
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, 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 being non-Lipschitz functions at their ending time, they don´t stand uniqueness 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. A valuable but little-known work on the subject is that of Houtain (1854).
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
Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.
Reduction to quadratures
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 complex differential equations require
complex numbers. 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 no longer 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.
Two memoirs by
Fuchs inspired a novel approach, subsequently elaborated by Thomé and
Frobenius. Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868.
Clebsch (1873) attacked the theory along lines parallel to those in his theory of
Abelian integrals. 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 f = 0 under rational one-to-one transformations.
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. 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.
Sturm–Liouville theory is a theory of a special type of second order linear ordinary differential equation. Their solutions are based on
eigenvalues and corresponding
eigenfunctions of linear operators defined via second-order
homogeneous linear equations. The problems are identified as Sturm-Liouville Problems (SLP) and are named after
J.C.F. Sturm and
J. Liouville, who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and the corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. SLPs are also useful in the analysis of certain partial differential equations.
Existence and uniqueness of solutions
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
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 that take the form F(x, y), and it can also be applied to systems of equations.
Global uniqueness and maximum domain of solution
When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:
For each initial condition (x0, y0) 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 .
In the case that , there are exactly two possibilities
explosion in finite time:
leaves domain of definition:
where Ω is the open set in which F is defined, and is its boundary.
Note that the maximum domain of the solution
is always an interval (to have uniqueness)
may be smaller than
may depend on the specific choice of (x0, y0).
This means that F(x, y) = y2, which is C1 and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.
Even in such a simple setting, the maximum domain of solution cannot be all 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 because
which is one of the two possible cases according to the above theorem.
Reduction of order
Differential equations can usually be solved more easily if the order of the equation can be reduced.
Reduction to a first-order system
Any explicit 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:
Summary of exact solutions
Some differential equations have solutions that 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 that 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).
First-order, separable in x and y (general case, see below for special cases)
When all other methods for solving an ODE fail, or in the cases where we have some intuition about what the solution to a DE might look like, it is sometimes possible to solve a DE simply by guessing the solution and validating it is correct. To use this method, we simply guess a solution to the differential equation, and then plug the solution into the differential equation to validate if it satisfies the equation. If it does then we have a particular solution to the DE, otherwise we start over again and try another guess. For instance we could guess that the solution to a DE has the form: since this is a very common solution that physically behaves in a sinusoidal way.
In the case of a first order ODE that is non-homogeneous we need to first find a DE solution to the homogeneous portion of the DE, otherwise known as the characteristic equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions together to obtain the total solution to the ODE, that is: