In mathematics, an ordinary differential equation (ODE) is a
differential equation (DE) dependent on only a single independent
variable. As with any other DE, its unknown(s) consists of one (or more)
function(s) and involves the
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s of those functions.
The term "ordinary" is used in contrast with
''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with
''stochastic'' differential equations (SDEs) where the progression is random.
Differential equations
A
linear differential equation
In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form
a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x)
wher ...
is a differential equation that is defined by a
linear polynomial
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer ...
in the unknown function and its derivatives, that is an
equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
of the form
:
where
and
are arbitrary
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s that do not need to be linear, and
are the successive derivatives of the unknown function
of the variable
.
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most
elementary and
special functions that are encountered in
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
are solutions of linear differential equations (see
Holonomic function
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable d ...
). 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
Riccati equation
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form
y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2( ...
).
Some ODEs can be solved explicitly in terms of known functions and
integrals
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...
. When that is not possible, the equation for computing the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the solutions may be useful. For applied problems,
numerical methods for ordinary differential equations
Numerical methods for ordinary differential equations are methods used to find Numerical analysis, numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although ...
can supply an approximation of the solution.
Background
Ordinary differential equations (ODEs) arise in many contexts of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
social
Social organisms, including human(s), live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary or not.
Etymology
The word "social" derives fro ...
and
natural sciences
Natural science or empirical science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer ...
. 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.
Specific mathematical fields include
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
. Scientific fields include much of
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
(celestial mechanics),
meteorology
Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...
(weather modeling),
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
(reaction rates),
biology
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
(infectious diseases, genetic variation),
ecology
Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ...
and
population modeling (population competition),
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
(stock trends, interest rates and the market equilibrium price changes).
Many mathematicians have studied differential equations and contributed to the field, including
Newton,
Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...
, the
Bernoulli family
The Bernoulli family ( ; ; ) of Basel was a Patrician (post-Roman Europe), patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physic ...
,
Riccati,
Clairaut,
d'Alembert
Jean-Baptiste le Rond d'Alembert ( ; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanics, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''E ...
, and
Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
.
A simple example is
Newton's second law
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body re ...
of motion—the relationship between the displacement
and the time
of an object under the force
, is given by the differential equation
:
which constrains the
motion of a particle of constant mass
. In general,
is a function of the position
of the particle at time
. The unknown function
appears on both sides of the differential equation, and is indicated in the notation
.
Definitions
In what follows,
is a
dependent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...
representing an unknown function
of the
independent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...
. The
notation for differentiation
In differential calculus, there is no single standard notation for differentiation. Instead, several notations for the derivative of a Function (mathematics), function or a dependent variable have been proposed by various mathematicians, includin ...
varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the
Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ...
is more useful for differentiation and
integration, whereas
Lagrange's notation
is more useful for representing
higher-order derivatives compactly, and
Newton's notation is often used in physics for representing derivatives of low order with respect to time.
General definition
Given
, a function of
,
, and derivatives of
. Then an equation of the form
:
is called an ''
explicit
Explicit refers to something that is specific, clear, or detailed. It can also mean:
* Explicit knowledge, knowledge that can be readily articulated, codified and transmitted to others
* Explicit (text), the final words of a text; contrast with inc ...
ordinary differential equation of order
''.
More generally, an ''implicit'' ordinary differential equation of order
takes the form:
:
There are further classifications:
System of ODEs
A number of coupled differential equations form a system of equations. If
is a vector whose elements are functions;
vector-valued function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
of
\mathbf and its derivatives, then
:
\mathbf^ = \mathbf\left(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right)
is an ''explicit system of ordinary differential equations'' of ''order''
n and ''dimension''
m. In
column vector
In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some , c ...
form:
:
\begin
y_1^ \\
y_2^ \\
\vdots \\
y_m^
\end =
\begin
f_1 \left (x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right ) \\
f_2 \left (x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right ) \\
\vdots \\
f_m \left (x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right)
\end
These are not necessarily linear. The ''implicit'' analogue is:
:
\mathbf \left(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^ \right) = \boldsymbol
where
\boldsymbol=(0,0,\ldots,0) is the
zero vector
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An '' additive id ...
. In matrix form
:
\begin
f_1(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^) \\
f_2(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^) \\
\vdots \\
f_m(x,\mathbf,\mathbf',\mathbf'',\ldots, \mathbf^)
\end=\begin
0\\
0\\
\vdots\\
0
\end
For a system of the form
\mathbf \left(x,\mathbf,\mathbf'\right) = \boldsymbol, some sources also require that the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
\frac be
non-singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular or sounder, a group of boar, see List of animal names
* Singular (band), a Thai jazz pop duo
*'' Singular ...
in order to call this an implicit ODE
ystem an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed
differential algebraic equation
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.
The set of the solutions of such a system is a ...
s (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.
Presumably for additional derivatives, the
Hessian matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a functio ...
and so forth are also assumed non-singular according to this scheme, although note that
any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.
The behavior of a system of ODEs can be visualized through the use of a
phase portrait.
Solutions
Given a differential equation
:
F\left(x, y, y', \ldots, y^ \right) = 0
a function
u:I\subset\mathbb\to\mathbb, where
I is an interval, is called a ''solution'' or
integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpre ...
for
F, if
u is
n-times differentiable on
I, and
:
F(x,u,u',\ \ldots,\ u^)=0 \quad x \in I.
Given two solutions
u:J\subset\mathbb\to\mathbb and
v:I\subset\mathbb\to\mathbb,
u is called an ''extension'' of
v if
I\subset J and
:
u(x) = v(x) \quad x \in I.\,
A solution that has no extension is called a ''maximal solution''. A solution defined on all of
\mathbb is called a ''global solution''.
A ''general solution'' of an
nth-order equation is a solution containing
n arbitrary independent
constants of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the Set (mathematics), set of all antiderivatives of f(x) ...
. A ''particular solution'' is derived from the general solution by setting the constants to particular values, often chosen to fulfill set '
initial conditions
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). Fo ...
or
boundary conditions
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
'. 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
In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but inst ...
and
variation of parameters
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous differential equation, inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differenti ...
.
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 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:
:
y'= -\text(y)\sqrt,\,\,y(0)=1
Admits the finite duration solution:
:
y(x)=\frac\left(1-\frac+\left, 1-\frac\\right)^2
Theories
Singular solutions
The theory of
singular solutions of ordinary and
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s 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
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.
Reduction to quadratures
The primitive attempt in dealing with differential equations had in view a reduction to
quadratures, that is, expressing the solutions in terms of known function and their integrals. This is possible for linear equations with constant coefficients, it appeared in the 19th century that this is generally impossible in other cases. Hence, analysts began the study (for their own) of functions that are solutions of differential equations, thus opening a new and fertile field.
Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by quadratures, but whether a given differential equation suffices for the definition of a function, and, if so, what are the characteristic properties of such functions.
Fuchsian theory
Two memoirs by
Fuchs inspired a novel approach, subsequently elaborated by Thomé and
Frobenius Frobenius is a surname. Notable people with the surname include:
* Ferdinand Georg Frobenius (1849–1917), mathematician
** Frobenius algebra
** Frobenius endomorphism
** Frobenius inner product
** Frobenius norm
** Frobenius method
** Frobenius g ...
. 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.
Lie's theory
From 1870,
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial cont ...
'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 group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, be referred to a common source, and that ordinary differential equations that admit the same
infinitesimal transformation
In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 ...
s 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.
A general solution approach uses the symmetry property of differential equations, the continuous
infinitesimal transformation
In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 ...
s of solutions to solutions (
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, ...
). Continuous
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
,
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
, and
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
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.
Sturm–Liouville theory
Sturm–Liouville theory is a theory of a special type of second-order linear ordinary differential equation. Their solutions are based on
eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
and corresponding
eigenfunctions
In mathematics, an eigenfunction of a linear map, linear operator ''D'' defined on some function space is any non-zero function (mathematics), function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor calle ...
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 problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ...
s involving ODEs both locally and globally. The two main theorems are
:
In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of
Grönwall's inequality are met.
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.
Local existence and uniqueness theorem simplified
The theorem can be stated simply as follows.
[Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, ] For the equation and initial value problem:
y' = F(x,y)\,,\quad y_0 = y(x_0)
if
F and
\partial F/\partial y are continuous in a closed rectangle
R = _0-a,x_0+a\times _0-b,y_0+b/math>
in the x-y plane, where a and b are real (symbolically: a,b\in\mathbb) and
x denotes the Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
, square brackets denote
closed intervals, then there is an interval
I = _0-h,x_0+h\subset _0-a,x_0+a/math>
for some h\in\mathbb 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:
[Boscain; Chitour 2011, p. 21]
For each initial condition
(x_0,y_0) there exists a unique maximum (possibly infinite) open interval
:
I_ = (x_-,x_+), x_\pm \in \R \cup \, x_0 \in I_
such that any solution that satisfies this initial condition is a
restriction of the solution that satisfies this initial condition with domain
I_\max.
In the case that
x_\pm \neq \pm\infty, there are exactly two possibilities
*explosion in finite time:
\limsup_ \, y(x)\, \to \infty
*leaves domain of definition:
\lim_ y(x)\ \in \partial \bar
where
\Omega is the open set in which
F is defined, and
\partial \bar is its boundary.
Note that the maximum domain of the solution
* is always an interval (to have uniqueness)
* may be smaller than
\R
* may depend on the specific choice of
(x_0,y_0).
;Example.
:
y' = y^2
This means that
F(x,y)=y^2, which is
C^1 and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.
Even in such a simple setting, the maximum domain of solution cannot be all
\R since the solution is
:
y(x) = \frac
which has maximum domain:
:
\begin\R & y_0 = 0 \\ pt\left (-\infty, x_0+\frac \right ) & y_0 > 0 \\ pt\left (x_0+\frac,+\infty \right ) & y_0 < 0 \end
This shows clearly that the maximum interval may depend on the initial conditions. The domain of
y could be taken as being
\R \setminus (x_0+ 1/y_0), 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
:
\lim_ \, y(x)\, \to \infty,
which is one of the two possible cases according to the above theorem.
Reduction of order
Differential equations are usually easier to solve if the
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
of the equation can be reduced.
Reduction to a first-order system
Any explicit differential equation of order
n,
:
F\left(x, y, y', y'',\ \ldots,\ y^\right) = y^
can be written as a system of
n first-order differential equations by defining a new family of unknown functions
:
y_i = y^.\!
for
i=1,2,\ldots,n. The
n-dimensional system of first-order coupled differential equations is then
:
\begin
y_1'&=&y_2\\
y_2'&=&y_3\\
&\vdots&\\
y_'&=&y_n\\
y_n'&=&F(x,y_1,\ldots,y_n).
\end
more compactly in vector notation:
:
\mathbf'=\mathbf(x,\mathbf)
where
:
\mathbf=(y_1,\ldots,y_n),\quad \mathbf(x,y_1,\ldots,y_n)=(y_2,\ldots,y_n,F(x,y_1,\ldots,y_n)).
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;
b and
c are real given constants;
C_1,C_2,\ldots are arbitrary constants (
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
in general). The differential equations are in their equivalent and alternative forms that lead to the solution through integration.
In the integral solutions,
\lambda and
\varepsilon are dummy variables of integration (the continuum analogues of indices in
summation
In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, pol ...
), and the notation
\int^x F(\lambda)\, d\lambda just means to integrate
F(\lambda) with respect to
\lambda, then ''after'' the integration substitute
\lambda=x, without adding constants (explicitly stated).
Separable equations
General first-order equations
General second-order equations
Linear to the nth order equations
The guessing method
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:
y = Ae^ 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 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. Finally, we add both of these solutions together to obtain the general solution to the ODE, that is:
\text = \text + \text
Software for ODE solving
*
Maxima, an open-source
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
.
*
COPASI, a free (
Artistic License 2.0) software package for the integration and analysis of ODEs.
*
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
, a technical computing application (MATrix LABoratory)
*
GNU Octave
GNU Octave is a scientific programming language for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly ...
, a high-level language, primarily intended for numerical computations.
*
Scilab
Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simul ...
, an open source application for numerical computation.
*
Maple
''Acer'' is a genus of trees and shrubs commonly known as maples. The genus is placed in the soapberry family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated si ...
, a proprietary application for symbolic calculations.
*
Mathematica
Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
, a proprietary application primarily intended for symbolic calculations.
*
SymPy, a Python package that can solve ODEs symbolically
*
Julia (programming language)
Julia is a high-level programming language, high-level, general-purpose programming language, general-purpose dynamic programming language, dynamic programming language, designed to be fast and productive, for e.g. data science, artificial intel ...
, a high-level language primarily intended for numerical computations.
*
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, group theory, differentia ...
, an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics.
*
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, fast Fourier ...
, a Python package that includes an ODE integration module.
*
Chebfun, an open-source package, written in
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
, for computing with functions to 15-digit accuracy.
*
GNU R, an open source computational environment primarily intended for statistics, which includes packages for ODE solving.
See also
*
Boundary value problem
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
*
Examples of differential equations
*
Laplace transform applied to differential equations
*
List of dynamical systems and differential equations topics
*
Matrix differential equation
*
Method of undetermined coefficients
In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but inst ...
*
Recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
Notes
References
*
*
* .
* Polyanin, A. D. and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations'' (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003.
*
*
*
*
*
Bibliography
*
*
* W. Johnson
''A Treatise on Ordinary and Partial Differential Equations'' John Wiley and Sons, 1913, i
University of Michigan Historical Math Collection*
*
Witold Hurewicz, ''Lectures on Ordinary Differential Equations'', Dover Publications,
*.
*
*
A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, ''Handbook of First Order Partial Differential Equations'', Taylor & Francis, London, 2002.
* D. Zwillinger, ''Handbook of Differential Equations (3rd edition)'', Academic Press, Boston, 1997.
External links
*
EqWorld: The World of Mathematical Equations containing a list of ordinary differential equations with their solutions.
Online Notes / Differential Equationsby Paul Dawkins,
Lamar University
Lamar University (Lamar or LU) is a public university in Beaumont, Texas, United States. Lamar has been a member of the Texas State University System since 1995. It was the flagship institution of the former Lamar University System. As of the ...
.
Differential Equations S.O.S. Mathematics.
A primer on analytical solution of differential equationsfrom the Holistic Numerical Methods Institute, University of South Florida.
Ordinary Differential Equations and Dynamical Systemslecture notes by
Gerald Teschl.
Notes on Diffy Qs: Differential Equations for EngineersAn introductory textbook on differential equations by Jiri Lebl of
UIUC
The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United States. Established in 1867, it is the f ...
.
Modeling with ODEs using ScilabA tutorial on how to model a physical system described by ODE using Scilab standard programming language by Openeering team.
Solving an ordinary differential equation in Wolfram, Alpha
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Differential calculus