View source for Ordinary differential equation

{{Short description|Differential equation containing derivatives with respect to only one variable}}
[[Image:Parabolic trajectory.svg|right|thumb|250px|alt=parabolic projectile motion showing velocity vector|The [[Physics:Trajectory|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.]]
{{Differential equations}}

In [[Mathematics|mathematics]], an '''ordinary differential equation''' ('''ODE''') is a [[Differential equation|differential equation]] (DE) dependent on only a single independent [[Variable (mathematics)|variable]]. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the [[Derivative|derivative]]s of those functions.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&q=%22ordinary+differential%22|date=15 March 2012|publisher=Cengage Learning|isbn=978-1-285-40110-2|access-date=11 July 2019|archive-url=https://web.archive.org/web/20200117230630/https://books.google.com/books?id=pasKAAAAQBAJ&printsec=frontcover#v=snippet&q=%22ordinary%20differential%22&f=false|archive-date=17 January 2020|url-status=live}}</ref> The term "ordinary" is used in contrast with [[Partial differential equation|''partial'' differential equations]] which may be with respect to {{em|more than}} one independent variable.<ref>{{cite web|url=http://hsm.stackexchange.com/a/5032/1772|title=What is the origin of the term "ordinary differential equations"? |access-date=2016-07-28|website=hsm.stackexchange.com |publisher=[[Company:Stack Exchange|Stack Exchange]] }}</ref>

== Differential equations ==
A [[Linear differential equation|linear differential equation]] is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an [[Equation|equation]] of the form 
:<math>a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^{(n)}+b(x)=0,</math>
where {{tmath|a_0(x)}}, ..., {{tmath|a_n(x)}} and {{tmath|b(x)}} are arbitrary [[Differentiable function|differentiable function]]s that do not need to be linear, and  {{tmath|y', \ldots, y^{(n)} }} are the successive derivatives of the unknown function {{mvar|y}} of the variable {{mvar|x}}.

Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most [[Elementary function|elementary]] and [[Special functions|special]] functions that are encountered in [[Physics:Physics|physics]] and [[Applied mathematics|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 [[Riccati equation]]).

Some ODEs can be solved explicitly in terms of known functions and [[Antiderivative|integrals]]. When that is not possible, the equation for computing the [[Taylor series]] of the solutions may be useful. For applied problems, [[Numerical methods for ordinary differential equations|numerical methods for ordinary differential equations]] can supply an approximation of the solution.

==Background==

Ordinary differential equations (ODEs) arise in many contexts of mathematics and [[Social:Social science|social]] and [[Natural science|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.{{cn|date=June 2023}}

Specific mathematical fields include [[Geometry|geometry]] and [[Physics:Analytical mechanics|analytical mechanics]]. Scientific fields include much of [[Physics:Physics|physics]] and [[Astronomy:Astronomy|astronomy]] (celestial mechanics), [[Physics:Meteorology|meteorology]] (weather modeling), [[Chemistry:Chemistry|chemistry]] (reaction rates),<ref>Mathematics for Chemists, D.M. Hirst, Macmillan Press, 1976, (No ISBN) SBN: 333-18172-7</ref> [[Biology:Biology|biology]] (infectious diseases, genetic variation), [[Earth:Ecology|ecology]] and population modeling (population competition), [[Finance:Economics|economics]] (stock trends, interest rates and the market equilibrium price changes).

Many mathematicians have studied differential equations and contributed to the field, including [[Biography:Isaac Newton|Newton]], Leibniz, the [[Bernoulli family]], Riccati,  Clairaut, d'Alembert, and Euler.

A simple example is Newton's second law of motion&mdash;the relationship between the displacement ''x'' and the time ''t'' of an object under the force ''F'', is given by the differential equation

:<math>m \frac{\mathrm{d}^2 x(t)}{\mathrm{d}t^2} = F(x(t))\,</math>

which constrains the [[Physics:Projectile motion|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'')).<ref>{{harvtxt|Kreyszig|1972|p=64}}</ref><ref>{{harvtxt|Simmons|1972|pp=1,2}}</ref><ref>{{harvtxt|Halliday|Resnick|1977|p=78}}</ref><ref>{{harvtxt|Tipler|1991|pp=78–83}}</ref>

==Definitions==
In what follows,  ''y'' is a dependent variable representing an unknown function {{nowrap|1=''y'' = ''f''(''x'')}} of the independent variable ''x''. The [[Notation for differentiation|notation for differentiation]] varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the [[Notation for differentiation#Leibniz's notation|Leibniz's notation]] {{nowrap|({{sfrac|''dy''|''dx''}}, {{sfrac|''d''<sup>2</sup>''y''|''dx''<sup>2</sup>}}, …, {{sfrac|''d''<sup>''n''</sup>''y''|''dx''<sup>''n''</sup>}})}} is more useful for differentiation and integration, whereas [[Notation for differentiation#Lagrange's notation|Lagrange's notation]] {{nowrap|(''y''′, ''y''′′, …, ''y''<sup>(''n'')</sup>)}} is more useful for representing higher-order derivatives compactly, and [[Notation for differentiation#Newton's notation|Newton's notation]] <math>(\dot y, \ddot y, \overset{...}{y})</math> is often used in physics for representing derivatives of low order with respect to time.

===General definition===

Given ''F'', a function of ''x'', ''y'', and derivatives of ''y''. Then an equation of the form

:<math>F\left (x,y,y',\ldots, y^{(n-1)} \right )=y^{(n)}</math>

is called an ''explicit ordinary differential equation of order n''.<ref name="Harper 1976 127">{{harvtxt|Harper|1976|p=127}}</ref><ref>{{harvtxt|Kreyszig|1972|p=2}}</ref>

More generally, an ''implicit'' ordinary differential equation of order ''n'' takes the form:<ref>{{harvtxt|Simmons|1972|p=3}}</ref>

:<math>F\left(x, y, y', y'',\ \ldots,\ y^{(n)}\right) = 0</math>

There are further classifications:
{{glossary}}
{{term|[[Autonomous system (mathematics)|Autonomous]]}}{{defn|A differential is ''autonomous'' if it does not depend on the variable {{math|''x''}}.}}
{{term|[[Linear differential equation|Linear]]}}{{defn|
A differential equation is ''linear'' if <math>F</math> can be written as a [[Linear combination|linear combination]] of the derivatives of {{math|''y''}}; that is,  if can be rewritten as 
:<math>y^{(n)} = \sum_{i=0}^{n - 1} a_i(x) y^{(i)} + r(x)</math>
where {{math|''a''{{hairsp}}{{sub|''i''}}{{thinsp}}(''x'')}} and {{math|''r''{{hairsp}}(''x'')}} are continuous functions of {{mvar|x}}.<ref name="Harper 1976 127"/><ref name="Kreyszig 1972 24">{{harvtxt|Kreyszig|1972|p=24}}</ref><ref>{{harvtxt|Simmons|1972|p=47}}</ref>

The function ''r''(''x'') is called the ''source term'', leading to further classification.<ref name="Kreyszig 1972 24"/><ref>{{harvtxt|Harper|1976|p=128}}</ref>}}
{{glossary}}
{{term|[[Homogeneous differential equation|Homogeneous]]}}{{defn|A linear differential equation is ''homogeneous'' if {{math|''r''(''x'') {{=}} 0}}. In this case, there is always the "trivial solution" {{math|''y'' {{=}} 0}}.}}
{{term|[[Homogeneous differential equation#Homogeneous linear differential equations|Nonhomogeneous (or inhomogeneous)]]}}{{defn|A linear differential equation is  ''nonhomogeneous'' if {{math|''r''(''x'') ≠ 0}}. }}
{{glossary end}}
{{term|Non-linear}}{{defn|A differential equation that is not linear.}}
{{glossary end}}

===System of ODEs===
{{Main|System of differential equations}}
A number of coupled differential equations form a system of equations. If '''y''' is a vector whose elements are functions; '''y'''(''x'') = [''y''<sub>1</sub>(''x''), ''y''<sub>2</sub>(''x''),..., ''y<sub>m</sub>''(''x'')], and '''F''' is a [[Vector-valued function|vector-valued function]] of '''y''' and its derivatives, then

:<math>\mathbf{y}^{(n)} = \mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n-1)} \right)</math>

is an ''explicit system of ordinary differential equations'' of ''order'' ''n'' and ''dimension'' ''m''. In column vector form:

:<math>\begin{pmatrix}
y_1^{(n)} \\
y_2^{(n)} \\
\vdots \\
y_m^{(n)}
\end{pmatrix} =

\begin{pmatrix}
f_1 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n-1)} \right ) \\
f_2 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n-1)} \right ) \\
\vdots \\
f_m \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n-1)} \right)
\end{pmatrix}</math>

These are not necessarily linear. The ''implicit'' analogue is:

:<math>\mathbf{F} \left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n)} \right) = \boldsymbol{0}</math>

where '''0''' = (0, 0, ..., 0) is the zero vector. In matrix form

:<math>\begin{pmatrix}
f_1(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n)}) \\
f_2(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n)}) \\
\vdots \\
f_m(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n)})
\end{pmatrix}=\begin{pmatrix}
0\\
0\\
\vdots\\
0
\end{pmatrix}</math>

For a system of the form <math>\mathbf{F} \left(x,\mathbf{y},\mathbf{y}'\right) = \boldsymbol{0}</math>, some sources also require that the Jacobian matrix <math>\frac{\partial\mathbf{F}(x,\mathbf{u},\mathbf{v})}{\partial \mathbf{v}}</math> be [[Singular matrix|non-singular]] in order to call this an implicit ODE [system]; 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 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.<ref>{{harvtxt|Kreyszig|1972|p=12}}</ref><ref>{{harvtxt|Ascher|1998|p=12}}</ref><ref name="IlchmannReis2014">{{cite book|author1=Achim Ilchmann|author2=Timo Reis|title=Surveys in Differential-Algebraic Equations II|year=2014|publisher=Springer|isbn=978-3-319-11050-9|pages=104–105}}</ref> Presumably for additional derivatives, the [[Hessian matrix]] and so forth are also assumed non-singular according to this scheme,{{citation needed|date=December 2014}} although note that any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order,<ref>{{harvtxt|Ascher|1998|p=5}}</ref> 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|phase portrait]].

=== Solutions ===
Given a differential equation
:<math>F\left(x, y, y', \ldots, y^{(n)} \right) = 0</math>
a function {{nowrap|''u'': ''I'' ⊂ '''R''' → '''R'''}}, where ''I'' is an interval, is called a ''solution'' or [[Integral curve|integral curve]] for ''F'', if ''u'' is ''n''-times differentiable on ''I'', and
:<math>F(x,u,u',\ \ldots,\ u^{(n)})=0 \quad x \in I.</math>

Given two solutions {{nowrap|''u'': ''J'' ⊂ '''R''' → '''R'''}} and {{nowrap|''v'': ''I'' ⊂ '''R''' → '''R'''}}, ''u'' is called an ''extension'' of ''v'' if {{nowrap|''I'' ⊂ ''J''}} and
:<math>u(x) = v(x) \quad x \in I.\,</math>
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 ''n''th-order equation is a solution containing ''n'' arbitrary independent [[Constant of integration|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 value problem|initial conditions]] or [[Boundary value problem|boundary conditions]]'.<ref>{{harvtxt|Kreyszig|1972|p=78}}</ref>  A [[Singular solution|singular solution]] is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.<ref>{{harvtxt|Kreyszig|1972|p=4}}</ref>

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 [[Ordinary differential equation#The guessing method|guessing method]] section in this article, and is frequently used when discussing the [[Method of undetermined coefficients|method of undetermined coefficients]] and [[Variation of parameters|variation of parameters]].

=== Solutions of finite duration ===

For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration,<ref>{{cite book |author = Vardia T. Haimo |title = 1985 24th IEEE Conference on Decision and Control |chapter = Finite Time Differential Equations |year = 1985 |pages = 1729–1733 |doi = 10.1109/CDC.1985.268832 |s2cid = 45426376 |chapter-url=https://ieeexplore.ieee.org/document/4048613}}</ref> 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:
:<math>y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1</math>
Admits the finite duration solution:
:<math>y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2</math>

==Theories==

===Singular solutions===
The theory of [[Singular solution|singular solution]]s of ordinary and [[Partial differential equation|partial differential equation]]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). [[Biography:Jean Gaston Darboux|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 [[Biography:Felice Casorati (mathematician)|Casorati]] and [[Biography:Arthur Cayley|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 [[Quadrature (mathematics)|quadrature]]s. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the ''n''th degree, so it was the hope of analysts to find a general method for integrating any differential equation. [[Biography:Carl Friedrich Gauss|Gauss]] (1799) showed, however, that complex differential equations require [[Complex number|complex number]]s. 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.

===Fuchsian theory===
{{main article|Frobenius method}}
Two memoirs by Fuchs<ref>''Crelle'', 1866, 1868</ref> inspired a novel approach, subsequently elaborated by Thomé and [[Biography:Ferdinand Georg Frobenius|Frobenius]]. Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868. [[Biography:Alfred Clebsch|Clebsch]] (1873) attacked the theory along lines parallel to those in his theory of [[Abelian integral]]s. 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, [[Biography:Sophus Lie|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 group]]s, be referred to a common source, and that ordinary differential equations that admit the same [[Infinitesimal transformation|infinitesimal transformation]]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.<ref>{{harvtxt|Lawrence|1999|p=9}}</ref>

A general solution approach uses the symmetry property of differential equations, the continuous [[Infinitesimal transformation|infinitesimal transformation]]s of solutions to solutions ([[Lie theory]]). Continuous [[Group theory|group theory]], Lie algebras, and [[Differential geometry|differential geometry]] are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its [[Lax pair]]s, 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 ===
{{main article|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 and corresponding [[Eigenfunction|eigenfunctions]] of linear operators defined via second-order [[System of linear equations|homogeneous linear equations]]. The problems are identified as Sturm-Liouville Problems (SLP) and are named after J.C.F. Sturm and [[Biography:Joseph Liouville|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.<ref>Logan, J. (2013). Applied mathematics (Fourth ed.).</ref> 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|initial value problem]]s involving ODEs both locally and globally. The two main theorems are

:{| class="wikitable"
|-
! Theorem
! Assumption
! Conclusion
|-
|[[Peano existence theorem]]
||F [[Continuous function|continuous]]
||local existence only
|-
|[[Picard–Lindelöf theorem]]
||F Lipschitz continuous
||local existence and uniqueness
|-
|}

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.<ref>{{harvtxt|Ascher|1998|p=13}}</ref>

===Local existence and uniqueness theorem simplified===

The theorem can be stated simply as follows.<ref name= "EDEBVP" >Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, {{isbn|0-471-83824-1}}</ref> For the equation and initial value problem:
<math display="block"> y' = F(x,y)\,,\quad y_0 = y(x_0)</math>
if ''F'' and ∂''F''/∂''y'' are continuous in a closed rectangle
<math display="block">R = [x_0-a,x_0+a] \times [y_0-b,y_0+b]</math>
in the ''x-y'' plane, where ''a'' and ''b'' are [[Real number|real]] (symbolically: {{math|''a'', ''b'' ∈ '''R'''}}) and {{math|×}} denotes the [[Cartesian product]], square brackets denote closed intervals, then there is an interval
<math display="block">I = [x_0-h,x_0+h] \subset [x_0-a,x_0+a]</math>
for some {{math|''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:<ref>Boscain; Chitour 2011, p.&nbsp;21</ref>

For each initial condition (''x''<sub>0</sub>, ''y''<sub>0</sub>) there exists a unique maximum (possibly infinite) open interval

:<math>I_{\max} = (x_-,x_+), x_\pm \in \R \cup \{\pm \infty\}, x_0 \in I_{\max}</math>

such that any solution that satisfies this initial condition is a [[Restriction (mathematics)|restriction]] of the solution that satisfies this initial condition with domain <math>I_\max</math>.

In the case that <math>x_\pm \neq \pm\infty</math>, there are exactly two possibilities

*explosion in finite time: <math>\limsup_{x \to x_\pm} \|y(x)\| \to \infty</math>
*leaves domain of definition: <math>\lim_{x \to x_\pm} y(x)\ \in \partial \bar{\Omega}</math>

where Ω is the open set in which ''F'' is defined, and <math>\partial \bar{\Omega}</math> is its boundary.

Note that the maximum domain of the solution

* is always an interval (to have uniqueness)
* may be smaller than <math>\R</math>
* may depend on the specific choice of (''x''<sub>0</sub>, ''y''<sub>0</sub>).

;Example.

:<math>y' = y^2</math>

This means that ''F''(''x, y'') = ''y''<sup>2</sup>, which is ''C''<sup>1</sup> and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.

Even in such a simple setting, the maximum domain of solution cannot be all <math>\R</math> since the solution is

:<math>y(x) = \frac{y_0}{(x_0-x)y_0+1}</math>

which has maximum domain:

:<math>\begin{cases}\R & y_0 = 0 \\[4pt] \left (-\infty, x_0+\frac{1}{y_0} \right ) & y_0 > 0 \\[4pt] \left (x_0+\frac{1}{y_0},+\infty \right ) & y_0 < 0 \end{cases}</math>

This shows clearly that the maximum interval may depend on the initial conditions. The domain of ''y'' could be taken as being <math>\R \setminus (x_0+ 1/y_0),</math> 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 <math>\R</math> because

:<math>\lim_{x \to x_\pm} \|y(x)\| \to \infty,</math>

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 [[Differential equation#Order|order]] of the equation can be reduced.

=== Reduction to a first-order system === <!-- Redirect [[Stiff equation]] links here -->
Any explicit differential equation of order ''n'',

:<math>F\left(x, y, y', y'',\ \ldots,\ y^{(n-1)}\right) = y^{(n)}</math>

can be written as a system of ''n'' first-order differential equations by defining a new family of unknown functions

:<math>y_i = y^{(i-1)}.\!</math>

for ''i'' = 1, 2, ..., ''n''. The ''n''-dimensional system of first-order coupled differential equations is then

:<math>\begin{array}{rcl}
  y_1'&=&y_2\\
  y_2'&=&y_3\\
  &\vdots&\\
  y_{n-1}'&=&y_n\\
  y_n'&=&F(x,y_1,\ldots,y_n).
\end{array}
</math>

more compactly in vector notation:

:<math>\mathbf{y}'=\mathbf{F}(x,\mathbf{y})</math>

where
:<math>\mathbf{y}=(y_1,\ldots,y_n),\quad \mathbf{F}(x,y_1,\ldots,y_n)=(y_2,\ldots,y_n,F(x,y_1,\ldots,y_n)).</math>

==Summary of exact solutions==
<!--Please leave the tables alone. If a table format is really disliked then instead of simply deleting it - expand into prose. For now they are there to summarize some common forms of equations and their solutions. Details are in the other articles. -->

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, {{math|''P''(''x'')}}, {{math|''Q''(''x'')}}, {{math|''P''(''y'')}}, {{math|''Q''(''y'')}}, and {{math|''M''(''x'',''y'')}}, {{math|''N''(''x'',''y'')}} are any integrable functions of {{math|''x''}}, {{math|''y''}}; {{math|''b''}} and {{math|''c''}} are real given constants; {{math|''C''<sub>1</sub>, ''C''<sub>2</sub>, ...}} are arbitrary constants ([[Complex number|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|summation]]), and the notation {{math|∫<sup>''x''</sup> ''F''(''λ'') ''dλ''}} just means to integrate {{math|''F''(''λ'')}} with respect to {{mvar|λ}}, then ''after'' the integration substitute {{math|1=''λ'' = ''x''}}, without adding constants (explicitly stated).

=== Separable equations ===
{| class="wikitable"
|-
! scope="col" | Differential equation
! scope="col" | Solution method
! scope="col" | General solution
|-
| First-order, separable in ''x'' and ''y'' (general case, see below for special cases)<ref name="MHFT">Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M. R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISC_2N 978-0-07-154855-7</ref>

<math>\begin{align}
  P_1(x)Q_1(y) + P_2(x)Q_2(y)\,\frac{dy}{dx} &= 0 \\
         P_1(x)Q_1(y)\,dx + P_2(x)Q_2(y)\,dy &= 0
\end{align}</math>
| Separation of variables (divide by ''P''<sub>2</sub>''Q''<sub>1</sub>).
| <math> \int^x \frac{P_1(\lambda)}{P_2(\lambda)}\,d\lambda + \int^y \frac{Q_2(\lambda)}{Q_1(\lambda)}\,d\lambda = C </math>
|-
| First-order, separable in ''x''<ref name= "EDEBVP" />
<math>\begin{align}
  \frac{dy}{dx} &= F(x) \\
             dy &= F(x) \, dx
\end{align}</math>
| Direct integration.
| <math>y= \int^x F(\lambda) \, d\lambda + C </math>
|-
| First-order, autonomous, separable in ''y''<ref name= "EDEBVP" />

<math>\begin{align}
  \frac{dy}{dx} &= F(y) \\
             dy &= F(y) \, dx
\end{align}</math>
| [[Separation of variables]] (divide by ''F'').
| <math>x=\int^y \frac{d\lambda}{F(\lambda)} + C</math>
|-
| First-order, separable in ''x'' and ''y''<ref name= "EDEBVP" />

<math>\begin{align}
  P(y)\frac{dy}{dx} + Q(x) &= 0 \\
       P(y)\,dy + Q(x)\,dx &= 0
\end{align}</math>
| Integrate throughout.
| <math>\int^y P(\lambda)\, d\lambda + \int^x Q(\lambda)\,d\lambda = C</math>
|}

=== General first-order equations ===
{| class="wikitable"
|-
! scope="col" | Differential equation
! scope="col" | Solution method
! scope="col" | General solution
|-
| First-order, homogeneous<ref name="EDEBVP" />

<math>\frac{dy}{dx} = F \left( \frac y x \right ) </math>
| Set ''y = ux'', then solve by separation of variables in ''u'' and ''x''.
| <math> \ln (Cx) = \int^{y/x} \frac{d\lambda}{F(\lambda) - \lambda} </math>
|-
| First-order, separable<ref name= "MHFT" />

<math>\begin{align}
  yM(xy) + xN(xy)\,\frac{dy}{dx} &= 0 \\
         yM(xy)\,dx + xN(xy)\,dy &= 0
\end{align}</math>

| Separation of variables (divide by ''xy'').
|
<math> \ln (Cx) = \int^{xy} \frac{N(\lambda)\,d\lambda}{\lambda [N(\lambda)-M(\lambda)] } </math>

If ''N'' = ''M'', the solution is ''xy'' = ''C''.
|-
| [[Exact differential equation|Exact differential]], first-order<ref name= "EDEBVP" />

<math>\begin{align}
  M(x,y) \frac{dy}{dx} + N(x,y) &= 0 \\
        M(x,y)\,dy + N(x,y)\,dx &= 0
\end{align}</math>

where <math> \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} </math>

| Integrate throughout.
| <math>\begin{align}
  F(x,y) &= \int^x M(\lambda,y)\,d\lambda + \int^y Y(\lambda)\,d\lambda \\
         &= \int^y N(x,\lambda)\,d\lambda + \int^x X(\lambda)\,d\lambda=C
\end{align}</math>
where
<math display="block">Y(y)= N(x,y)-\frac{\partial}{\partial y}\int^x M(\lambda,y)\,d\lambda</math> and
<math display="block">X(x)= M(x,y)-\frac{\partial}{\partial x}\int^y N(x,\lambda)\,d\lambda </math>
|-
| [[Inexact differential equation|Inexact differential]], first-order<ref name= "EDEBVP" />

<math>\begin{align}
  M(x,y) \frac{dy}{dx} + N(x,y) &= 0 \\
        M(x,y)\,dy + N(x,y)\,dx &= 0
\end{align}</math>

where <math> \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} </math>
| Integration factor {{math|''μ''(''x'', ''y'')}} satisfying

<math> \frac{\partial (\mu M)}{\partial y} = \frac{\partial (\mu N)}{\partial x} </math>
| If {{math|''μ''(''x'', ''y'')}} can be found in a suitable way, then

<math>\begin{align}
  F(x,y) = &\int^x \mu(\lambda,y)M(\lambda,y)\,d\lambda + \int^y Y(\lambda)\,d\lambda \\
         = &\int^y \mu(x,\lambda)N(x,\lambda)\,d\lambda + \int^x X(\lambda)\,d\lambda = C
\end{align}</math>

where <math display="block">Y(y)= N(x,y)-\frac{\partial}{\partial y}\int^x \mu (\lambda,y)M(\lambda,y)\,d\lambda</math>
and <math display="block">X(x)= M(x,y)-\frac{\partial}{\partial x}\int^y \mu (x,\lambda)N(x,\lambda)\,d\lambda</math>
|}

=== General second-order equations ===
{| class="wikitable"
|-
! scope="col" | Differential equation
! scope="col" | Solution method
! scope="col" | General solution
|-
| Second-order, autonomous<ref>Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978, {{isbn|0-7135-1594-5}}</ref>

<math>\frac{d^2y}{dx^2} = F(y) </math>
| Multiply both sides of equation by {{math|2''dy''/''dx''}}, substitute {{nowrap|<math>2 \frac{dy}{dx} \frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right)^2 </math>,}} then integrate twice.
| <math> x = \pm \int^y \frac{ d \lambda}{\sqrt{2 \int^\lambda F(\varepsilon) \, d \varepsilon + C_1}} + C_2 </math>
|}

=== Linear to the ''n''th order equations ===
{| class="wikitable"
|-
! scope="col" | Differential equation
! scope="col" | Solution method
! scope="col" | General solution
|-
| First-order, linear, inhomogeneous, function coefficients<ref name="EDEBVP" />

<math>\frac{dy}{dx} + P(x)y = Q(x)</math>

| Integrating factor: <math>e^{\int^x P(\lambda)\,d\lambda}.</math>
| Armour formula:
<math>y = e^{- \int^x P(\lambda) \, d\lambda}\left[\int^x e^{\int^\lambda P(\varepsilon) \, d\varepsilon}Q(\lambda) \, d\lambda +C \right]</math>
|-
|Second-order, linear, inhomogeneous, function coefficients
<math>\frac{d^2y}{dx^2}+2p(x)\frac{dy}{dx}+\left(p(x)^2+p'(x)\right)y=q(x) </math>
|Integrating factor: <math>e^{\int^x P(\lambda)\,d\lambda}</math>
|<math>y = e^{- \int^x P(\lambda) \, d\lambda}\left[\int^x\left(\int^\xi e^{\int^\lambda P(\varepsilon) \, d\varepsilon}Q(\lambda) \, d\lambda \right)d\xi +C_1x+C_2\right]</math>
|-
| Second-order, linear, inhomogeneous, constant coefficients<ref name="MMPE">Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISC_2N 978-0-521-86153-3</ref>

<math>\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = r(x)</math>
| Complementary function ''y<sub>c</sub>'': assume {{nowrap|1=''y<sub>c</sub>'' = ''e''<sup>α''x''</sup>,}} substitute and solve polynomial in α, to find the linearly independent functions <math>e^{\alpha_j x}</math>.

Particular integral ''y<sub>p</sub>'': in general the method of variation of parameters, though for very simple ''r''(''x'') inspection may work.<ref name="EDEBVP" />
| <math>y = y_c + y_p</math>

If {{math|''b''<sup>2</sup> > 4''c''}}, then

<math>y_c = C_1e^{ -\frac x2\,\left(b + \sqrt{b^2 - 4c}\right)} + C_2e^{-\frac x2\,\left(b - \sqrt{b^2 - 4c}\right)}</math>

If {{math|1=''b''<sup>2</sup> = 4''c''}}, then

<math>y_c = (C_1x + C_2)e^{-\frac{bx}{2}}</math>

If {{math|''b''<sup>2</sup> < 4''c''}}, then

<math>y_c = e^{ -\frac{bx}{2}} \left[ C_1 \sin\left( x\,\frac{\sqrt{4c-b^2}}{2}\right) + C_2\cos\left( x\,\frac{\sqrt{4c-b^2}}{2}\right)\right]</math>
|-
|''n''th-order, linear, inhomogeneous, constant coefficients<ref name="MMPE" />

<math> \sum_{j=0}^n b_j \frac{d^j y}{dx^j} = r(x)</math>
| Complementary function ''y<sub>c</sub>'': assume {{nowrap|1=''y<sub>c</sub>'' = ''e''<sup>α''x''</sup>,}} substitute and solve polynomial in α, to find the linearly independent functions <math>e^{\alpha_j x}</math>.

Particular integral ''y<sub>p</sub>'': in general the method of variation of parameters, though for very simple ''r''(''x'') inspection may work.<ref name="EDEBVP" />
|<math>y = y_c + y_p</math>

Since ''α''<sub>''j''</sub> are the solutions of the [[Polynomial|polynomial]] of [[Degree of a polynomial|degree]] ''n'': <math display="inline"> \prod_{j=1}^n ( \alpha - \alpha_j) = 0 </math>, then:
for ''α''<sub>''j''</sub> all different,
<math display="block"> y_c = \sum_{j=1}^n C_j e^{\alpha_j x} </math>
for each root ''α''<sub>''j''</sub> repeated ''k<sub>j</sub>'' times,
<math display="block"> y_c = \sum_{j=1}^n \left( \sum_{\ell=1}^{k_j} C_{j,\ell} x^{\ell-1}\right )e^{\alpha_j x}</math>
for some ''α''<sub>''j''</sub> complex, then setting ''α'' = ''χ''<sub>''j''</sub> + ''iγ''<sub>''j''</sub>, and using [[Euler's formula]], allows some terms in the previous results to be written in the form
<math display="block"> C_j e^{\alpha_j x} = C_j e^{\chi_j x}\cos(\gamma_j x + \varphi_j)</math>
where ''ϕ''<sub>''j''</sub> is an arbitrary constant (phase shift).
|}

==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: <math>y = Ae^{\alpha t}</math> 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:

<math>\text{total solution} = \text{homogeneous solution} + \text{particular solution}</math>

== Software for ODE solving ==
* [[Software:Maxima|Maxima]], an open-source [[Software:Computer algebra system|computer algebra system]].
* [[Software:COPASI|COPASI]], a free ([[Software:Artistic License|Artistic License 2.0]]) software package for the integration and analysis of ODEs.
* [[Software:MATLAB|MATLAB]], a technical computing application (MATrix LABoratory)
* [[Software:GNU Octave|GNU Octave]], a high-level language, primarily intended for numerical computations.
* [[Software:Scilab|Scilab]], an open source application for numerical computation.
* [[Software:Maple|Maple]], a proprietary application for symbolic calculations. 
* [[Software:Mathematica|Mathematica]], a proprietary application primarily intended for symbolic calculations. 
* [[Software:SymPy|SymPy]], a Python package that can solve ODEs symbolically
* [[Julia (programming language)]], a high-level language primarily intended for numerical computations.
* [[Software:SageMath|SageMath]], an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics.
* [[Software:SciPy|SciPy]], a Python package that includes an ODE integration module.
* [[Software:Chebfun|Chebfun]], an open-source package, written in [[Software:MATLAB|MATLAB]], 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]]
*[[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]]
*[[Recurrence relation]]

== Notes ==
{{reflist|2}}

==References==
* {{citation | first1 = David | last1 = Halliday |  first2 = Robert | last2 = Resnick | author-link2 = Robert Resnick | year = 1977 | isbn = 0-471-71716-9 | title = Physics | edition = 3rd | publisher = {{wipe|John Wiley & Sons}} | location = New York }}
* {{citation | first1 = Charlie | last1 = Harper | year = 1976 | isbn = 0-13-487538-9 | title = Introduction to Mathematical Physics | publisher = Prentice-Hall | location = New Jersey }}
* {{citation | first1 = Erwin | last1 = Kreyszig |  year = 1972 | isbn = 0-471-50728-8 | title = Advanced Engineering Mathematics | edition = 3rd | publisher = {{wipe|John Wiley & Sons}} | location = New York | url-access = registration | url = https://archive.org/details/advancedengineer00krey }}.
* Polyanin, A. D. and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations'' (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. {{isbn|1-58488-297-2}}
* {{citation | first1 = George F. | last1 = Simmons |  year = 1972 | title = Differential Equations with Applications and Historical Notes | publisher = McGraw-Hill | location = New York | lccn = 75173716 }}
* {{citation | first1 = Paul A. | last1 = Tipler | year = 1991 | isbn = 0-87901-432-6 | title = Physics for Scientists and Engineers: Extended version | edition = 3rd | publisher = Worth Publishers | location = New York }}
* {{citation | first1 = Ugo | last1 = Boscain | first2 = Yacine | last2 = Chitour | year = 2011 | title = Introduction à l'automatique | url = http://www.cmapx.polytechnique.fr/~boscain/poly2011.pdf | language = fr}}
* {{citation | first1 = Lawrence | last1 = Dresner | year = 1999 | title = Applications of Lie's Theory of Ordinary and Partial Differential Equations | isbn = 978-0750305303 | publisher = Institute of Physics Publishing | location = Bristol and Philadelphia }}
* {{citation | first1 = Uri | last1 = Ascher | first2 = Linda | last2 = Petzold | title = Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations | year = 1998 | publisher = SIAM | isbn = 978-1-61197-139-2}}

== Bibliography ==
* {{cite book | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | url=https://archive.org/details/theoryofordinary00codd | url-access=registration | publisher=McGraw-Hill | location=New York | year=1955}}
* {{Citation | last1=Hartman | first1=Philip |title=Ordinary differential equations | orig-year=1964 | url=https://books.google.com/books?id=CENAPMUEpfoC | publisher=Society for Industrial and Applied Mathematics | location=Philadelphia | series=Classics in Applied Mathematics | isbn=978-0-89871-510-1 |mr=1929104 | year=2002 | volume=38 | doi=10.1137/1.9780898719222}}
* W. Johnson, [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 ''A Treatise on Ordinary and Partial Differential Equations''], John Wiley and Sons, 1913, in [http://hti.umich.edu/u/umhistmath/ University of Michigan Historical Math Collection]
* {{Citation | last1=Ince | first1=Edward L. | title=Ordinary Differential Equations | orig-year=1926 | url=https://archive.org/details/ordinarydifferen029666mbp | publisher=Dover Publications, New York | isbn=978-0-486-60349-0 | mr=0010757  | year=1944}}
* Witold Hurewicz, ''Lectures on Ordinary Differential Equations'', Dover Publications, {{isbn|0-486-49510-8}}
*{{Cite book |first=Nail H. |last=Ibragimov |title=CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3 |publisher=CRC-Press |location=Providence |year=1993 |isbn=0-8493-4488-3 }}.
* {{cite book| last = Teschl| given = Gerald| title = Ordinary Differential Equations and Dynamical Systems| publisher=American Mathematical Society| place = Providence| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
* [[Biography:Andrei Polyanin|A. D. Polyanin]], V. F. Zaitsev, and A. Moussiaux, ''Handbook of First Order Partial Differential Equations'', Taylor & Francis, London, 2002. {{isbn|0-415-27267-X}}
* D. Zwillinger, ''Handbook of Differential Equations (3rd edition)'', Academic Press, Boston, 1997.

==External links==
*{{springer|title=Differential equation, ordinary|id=p/d031910}}
*[http://eqworld.ipmnet.ru/index.htm EqWorld: The World of Mathematical Equations], containing a list of ordinary differential equations with their solutions.
*[http://tutorial.math.lamar.edu/classes/de/de.aspx Online Notes / Differential Equations] by Paul Dawkins, [[Organization:Lamar University|Lamar University]].
*[http://www.sosmath.com/diffeq/diffeq.html Differential Equations], S.O.S. Mathematics.
*[http://numericalmethods.eng.usf.edu/mws/gen/08ode/mws_gen_ode_bck_primer.pdf A primer on analytical solution of differential equations] from the Holistic Numerical Methods Institute, University of South Florida.
*[https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ Ordinary Differential Equations and Dynamical Systems] lecture notes by [[Biography:Gerald Teschl|Gerald Teschl]].
*[http://www.jirka.org/diffyqs/ Notes on Diffy Qs: Differential Equations for Engineers] An introductory textbook on differential equations by Jiri Lebl of UIUC.
*[http://www.openeering.com/sites/default/files/LHY_Scilab_Tutorial_Part1.pdf Modeling with ODEs using Scilab] A tutorial on how to model a physical system described by ODE using Scilab standard programming language by Openeering team.
*[http://www.wolframalpha.com/input/?i=y%27%27+%2B+2xy+%3D+0 Solving an ordinary differential equation in Wolfram|Alpha]

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