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In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions.BOOK, Dennis G. Zill, A First Course in Differential Equations with Modeling Applications,weblink 15 March 2012, Cengage Learning, 1-285-40110-7, The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.WEB,weblink What is the origin of the term "ordinary differential equations"?, 2016-07-28, hsm.stackexchange.com, Stack Exchange,

## Differential equations

A 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 of the form
a_0(x)y +a_1(x)y' + a_2(x)y'' +cdots +a_n(x)y^{(n)}+b(x)=0,
where {{tmath|a_0(x)}}, ..., {{tmath|a_n(x)}} and {{tmath|b(x)}} are arbitrary differentiable functions 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 and special functions that are encountered in physics and 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 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 can supply an approximation of the solution.

## Background

Image:Parabolic trajectory.svg|right|thumb|250px|alt=parabolic projectile motion showing velocity vector|The trajectory of a projectile launched from a cannoncannonOrdinary differential equations (ODEs) arise in many contexts of mathematics and social 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.Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), meteorology (weather modelling), chemistry (reaction rates),Mathematics for Chemists, D.M. Hirst, Macmillan Press, 1976, (No ISBN) SBN: 333-18172-7 biology (infectious diseases, genetic variation), ecology and population modelling (population competition), economics (stock trends, interest rates and the market equilibrium price changes).Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert, and Euler.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
m frac{mathrm{d}^2 x(t)}{mathrm{d}t^2} = F(x(t)),
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)).{{harvtxt|Kreyszig|1972|p=64}}{{harvtxt|Simmons|1972|pp=1,2}}{{harvtxt|Halliday|Resnick|1977|p=78}}{{harvtxt|Tipler|1991|pp=78â€“83}}

## Definitions

In what follows, let y be a dependent variable and x an independent variable, and y = f(x) is an unknown function of x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the Leibniz's notation (dy/dx,d2y/dx2,...,d'n'y/dxn) is more useful for differentiation and integration, whereas Lagrange's notation (y′,y′′, ..., y(n)) is more useful for representing derivatives of any order compactly, and Newton's notation (dot y, ddot y, overset{...}{y}) 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
Fleft (x,y,y',ldots, y^{(n-1)} right )=y^{(n)}
is called an explicit ordinary differential equation of order n.{{harvtxt|Harper|1976|p=127}}{{harvtxt|Kreyszig|1972|p=2}}More generally, an implicit ordinary differential equation of order n takes the form:{{harvtxt|Simmons|1972|p=3}}
Fleft(x, y, y', y'', ldots, y^{(n)}right) = 0
There are further classifications:{{glossary}}{{term|Autonomous}}{{defn|A differential equation not depending on x is called autonomous.}}{{term|Linear}}{hide}defn|A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y:
y^{(n)} = sum_{i=0}^{n - 1} a_i(x) y^{(i)} + r(x)
where {{math|a{{hairsp{edih}{{sub|i}}{{thinsp}}(x)}} and {{math|r{{hairsp}}(x)}} are continuous functions of {{mvar|x}}.{{harvtxt|Kreyszig|1972|p=24}}{{harvtxt|Simmons|1972|p=47}}The function r(x) is called the source term, leading to two further important classifications:{{harvtxt|Harper|1976|p=128}}}}{{glossary}}{{term|Homogeneous}}{{defn|If r(x) {{=}} 0, and consequently one "automatic" solution is the trivial solution, y {{=}} 0. The solution of a linear homogeneous equation is a complementary function, denoted here by yc.{{term|Nonhomogeneous (or inhomogeneous)}}{{defn|If r(x) â‰  0. The additional solution to the complementary function is the particular integral, denoted here by yp.}}{{glossary end}}The general solution to a linear equation can be written as y {{=}} yc + yp.}}{{term|Non-linear}}{{defn|A differential equation that cannot be written in the form of a linear combination.}}{{glossary end}}

### System of ODEs

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
mathbf{y}^{(n)} = mathbf{F}left(x,mathbf{y},mathbf{y}',mathbf{y}'',ldots, mathbf{y}^{(n-1)} right)
is an explicit system of ordinary differential equations of order n and dimension m. In column vector form:
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}These are not necessarily linear. The implicit analogue is:
mathbf{F} left(x,mathbf{y},mathbf{y}',mathbf{y}'',ldots, mathbf{y}^{(n)} right) = boldsymbol{0}
where 0 = (0, 0, ..., 0) is the zero vector. In matrix form
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}vdotsend{pmatrix}For a system of the form mathbf{F} left(x,mathbf{y},mathbf{y}'right) = boldsymbol{0}, some sources also require that the Jacobian matrix frac{partialmathbf{F}(x,mathbf{u},mathbf{v})}{partial mathbf{v}} be 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.BOOK, Uri M. Ascher, Linda R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, 1998, SIAM, 978-1-61197-139-2, 12, BOOK, Achim Ilchmann, Timo Reis, Surveys in Differential-Algebraic Equations II, 2014, Springer, 978-3-319-11050-9, 104â€“105, 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,BOOK, Uri M. Ascher, Linda R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, 1998, SIAM, 978-1-61197-139-2, 5, 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
Fleft(x, y, y', ldots, y^{(n)} right) = 0
a function {{nowrap|u: I âŠ‚ R â†’ R}} is called a solution or integral curve for F, if u is n-times differentiable on I, and
F(x,u,u', ldots, u^{(n)})=0 quad x in I.
Given two solutions {{nowrap|u: J âŠ‚ R â†’ R}} and {{nowrap|v: I âŠ‚ R â†’ R}}, u is called an extension of v if {{nowrap|I âŠ‚ 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 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'.{{harvtxt|Kreyszig|1972|p=78}} A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.{{harvtxt|Kreyszig|1972|p=4}}

## Theories

### Singular solutions

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 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. 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.

### Fuchsian theory

Two memoirs by FuchsCrelle, 1866, 1868 inspired a novel approach, subsequently elaborated by ThomÃ© 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.

### Lie's theory

From 1870, Sophus Lie's work put the theory of differential equations on a better foundation. He showed that the integration theories of the older mathematicians can, using Lie groups, be referred to a common source, and that ordinary differential equations that admit the same infinitesimal transformations present comparable integration difficulties. 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.{{harvtxt|Lawrence|1999|p=9}}A general solution approach uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras, and differential geometry are used to understand the structure of linear and nonlinear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, BÃ¤cklund 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.

### Sturmâ€“Liouville theory

Sturmâ€“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.Logan, J. (2013). Applied mathematics (Fourth ed.). 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
{| class="wikitable"! Theorem! Assumption! Conclusion
|Peano existence theorem
F continuouslocal existence only
|Picardâ€“LindelÃ¶f theorem
F Lipschitz continuouslocal 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 GrÃ¶nwall'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.BOOK, Uri M. Ascher, Linda R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, 1998, SIAM, 978-1-61197-139-2, 13,

### 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, {{isbn|0-471-83824-1}} For the equation and initial value problem:
y' = F(x,y),,quad y_0 = y(x_0)
if F and âˆ‚F/âˆ‚y are continuous in a closed rectangle
R=[x_0-a,x_0+a]times [y_0-b,y_0+b]
in the x-y plane, where a and b are real (symbolically: a, b âˆˆ â„) and Ã— denotes the cartesian product, square brackets denote closed intervals, then there is an interval
I = [x_0-h,x_0+h] subset [x_0-a,x_0+a]
for some h âˆˆ â„ 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 Picardâ€“LindelÃ¶f theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:Boscain; Chitour 2011, p. 21For each initial condition (x0, y0) there exists a unique maximum (possibly infinite) open interval
I_{max} = (x_-,x_+), x_pm in R cup {pm infty}, x_0 in I_{max}
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 pminfty, there are exactly two possibilities
• explosion in finite time: limsup_{x to x_pm} |y(x)| to infty
• leaves domain of definition: lim_{x to x_pm} y(x) in partial bar{Omega}
where Î© is the open set in which F is defined, and partial bar{Omega} 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 (x0, y0).

Example.
y' = y^2
This means that F(x, y) = y2, which is C1 and therefore locally Lipschitz continuous, satisfying the Picardâ€“LindelÃ¶f theorem.Even in such a simple setting, the maximum domain of solution cannot be all R since the solution is
y(x) = frac{y_0}{(x_0-x)y_0+1}
which has maximum domain:
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}
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_{x to x_pm} |y(x)| to infty,
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,
Fleft(x, y, y', y'', ldots, y^{(n-1)}right) = y^{(n)}
can be written as a system of n first-order differential equations by defining a new family of unknown functions
y_i = y^{(i-1)}.!
for i = 1, 2,..., n. The n-dimensional system of first-order coupled differential equations is then
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}more compactly in vector notation:
mathbf{y}'=mathbf{F}(x,mathbf{y})
where

## 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, Î» and Îµ are dummy variables of integration (the continuum analogues of indices in summation), and the notation âˆ«xF(Î»dÎ» just means to integrate F(Î») with respect to Î», then after the integration substitute Î» = x, without adding constants (explicitly stated).
{| class="wikitable"! scope="col" | Type! scope="col" | Differential equation! scope="col" | Solution method! scope="col" | General solution
Separable| First-order, separable in x and y (general case, see below for special cases)Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M. R. Spiegel, J. Liu, Schuam's Outline Series, 2009, ISC_2N 978-0-07-154855-7begin{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}| Separation of variables (divide by P2Q1).| int^x frac{P_1(lambda)}{P_2(lambda)},dlambda + int^y frac{Q_2(lambda)}{Q_1(lambda)},dlambda = C ,!
| First-order, separable in xbegin{align}
frac{dy}{dx} &= F(x)
dy &= F(x) , dx
end{align}| Direct integration.| y= int^x F(lambda) , dlambda + C ,!
| First-order, autonomous, separable in ybegin{align}
frac{dy}{dx} &= F(y)
dy &= F(y) , dx
end{align}| Separation of variables (divide by F).| x=int^y frac{dlambda}{F(lambda)}+C,!
| First-order, separable in x and ybegin{align}
P(y)frac{dy}{dx} + Q(x) &= 0
P(y),dy + Q(x),dx &= 0
end{align}| Integrate throughout.| int^y P(lambda), dlambda + int^x Q(lambda),dlambda = C,!
General first-order| First-order, homogeneousfrac{dy}{dx} = F left( frac y x right ) ,!| Set y = ux, then solve by separation of variables in u and x.| ln (Cx) = int^{y/x} frac{dlambda}{F(lambda) - lambda} , !
| First-order, separablebegin{align}
yM(xy) + xN(xy),frac{dy}{dx} &= 0
yM(xy),dx + xN(xy),dy &= 0
end{align}| Separation of variables (divide by xy).|
ln (Cx) = int^{xy} frac{N(lambda),dlambda}{lambda [N(lambda)-M(lambda)] } ,!
If N = M, the solution is xy = C.
Exact differential equation>Exact differential, first-orderbegin{align}
M(x,y) frac{dy}{dx} + N(x,y) &= 0
M(x,y),dy + N(x,y),dx &= 0
end{align}where frac{partial M}{partial x} = frac{partial N}{partial y} , ! | Integrate throughout.| begin{align}
F(x,y) = &int^y M(x,lambda),dlambda + int^x N(lambda,y),dlambda
&+ Y(y) + X(x) = C
end{align}where Y(y) and X(x) are functions from the integrals rather than constant values, which are set to make the final function F(x, y) satisfy the initial equation.
Inexact differential equation>Inexact differential, first-orderbegin{align}
M(x,y) frac{dy}{dx} + N(x,y) &= 0
M(x,y),dy + N(x,y),dx &= 0
end{align}where frac{partial M}{partial x} neq frac{partial N}{partial y} , ! | Integration factor Î¼(x, y) satisfying
frac{partial (mu M)}{partial x} = frac{partial (mu N)}{partial y} , ! | If Î¼(x, y) can be found:
begin{align}
F(x,y) = &int^y mu(x,lambda)M(x,lambda),dlambda + int^x mu(lambda,y)N(lambda,y),dlambda
&+ Y(y) + X(x) = C
end{align}
|General second-order
0-7135-1594-5}}frac{d^2y}{dx^2} = F(y) ,!| Multiply both sides of equation by 2dy/dx, substitute 2 frac{dy}{dx}frac{d^2y}{dx^2} = frac{d}{dx}left(frac{dy}{dx}right)^2 ,!, then integrate twice.| x = pm int^y frac{ d lambda}{sqrt{2 int^lambda F(varepsilon) , d varepsilon + C_1}} + C_2 , !
Linear to nth order| First-order, linear, inhomogeneous, function coefficientsfrac{dy}{dx} + P(x)y = Q(x),!| Integrating factor: e^{int^x P(lambda),dlambda}.| y = e^{- int^x P(lambda) , dlambda}left[int^x e^{int^lambda P(varepsilon) , dvarepsilon}Q(lambda) , dlambda +C right]
| Second-order, linear, inhomogeneous, constant coefficientsMathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISC_2N 978-0-521-86153-3frac{d^2y}{dx^2} + bfrac{dy}{dx} + cy = r(x),!| Complementary function yc: assume yc = eÎ±x, substitute and solve polynomial in Î±, to find the linearly independent functions e^{alpha_j x}.Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.| y = y_c + y_pIf {{math|b2 > 4c}}, then
y_c = C_1e^{ -frac x2,left(b + sqrt{b^2 - 4c}right)} + C_2e^{-frac x2,left(b - sqrt{b^2 - 4c}right)}
If {{math|1=b2 = 4c}}, then
y_c = (C_1x + C_2)e^{-frac{bx}{2}}
If {{math|b2 < 4c}}, then
y_c = e^{ -frac{bx}{2}} left[ C_1 sinleft( x,frac{sqrt{4c-b^2}}{2}right) + C_2cosleft( x,frac{sqrt{4c-b^2}}{2}right)right]
| nth-order, linear, inhomogeneous, constant coefficients
sum_{j=0}^n b_j frac{d^j y}{dx^j} = r(x),!| Complementary function yc: assume yc = eÎ±x, substitute and solve polynomial in Î±, to find the linearly independent functions e^{alpha_j x}.
Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.| y = y_c + y_pSince Î±j are the solutions of the polynomial of degreen: prod_{j=1}^n ( alpha - alpha_j) = 0 ,!, then:for Î±j all different,
y_c = sum_{j=1}^n C_j e^{alpha_j x} ,!
for each root Î±j repeated kj times,
y_c = sum_{j=1}^n left( sum_{ell=1}^{k_j} C_{j,ell} x^{ell-1}right )e^{alpha_j x} ,!
for some Î±j complex, then setting Î± = Ï‡j + iÎ³j, and using Euler's formula, allows some terms in the previous results to be written in the form
C_je^{alpha_j x} = C_j e^{chi_j x}cos(gamma_j x + varphi_j),!
where Ï•j is an arbitrary constant (phase shift).

## Software for ODE solving

• Maxima, an open-source computer algebra system.
• COPASI, a free (Artistic License 2.0) software package for the integration and analysis of ODEs.
• MATLAB, a technical computing application (MATrix LABoratory)
• GNU Octave, a high-level language, primarily intended for numerical computations.
• Scilab, an open source application for numerical computation.
• Maple, a proprietary application for symbolic calculations.
• Mathematica, a proprietary application primarily intended for symbolic calculations.
• Julia (programming language), a high-level, multi-paradigm, open-source, dynamic programming language primarily intended for numerical computations, although it is flexible enough for general-purpose programming.
• SageMath, an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics.
• SciPy, a Python package that includes an ODE integration module.
• Chebfun, an open-source package, written in MATLAB, for computing with functions to 15-digit accuracy.
• GNU R, an open source computational environment primarily intended for statistics, which includes package for ODE solving.
• EROS.NET a free ODE solver for .NET.

{{reflist|2}}

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• 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 | authorlink = George F. 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 Ã  l'automatique | url =weblink | language = french}}
• {{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 }}

## Bibliography

• BOOK, Coddington, Earl A., Levinson, Norman, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955,
• {{Citation | last1=Hartman | first1=Philip |authorlink=Philip Hartman |title=Ordinary differential equations | origyear=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, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
• {{Citation | last1=Ince | first1=Edward L. |authorlink=Edward Lindsay Ince | title=Ordinary Differential Equations | origyear=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}}
• BOOK, Nail H, Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3, CRC-Press, Providence, 1993, 0-8493-4488-3, {{inconsistent citations, }}.
• BOOK, Teschl, Gerald, Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, Providence, Rhode Island, Providence, 2012, 978-0-8218-8328-0,weblink
• 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.

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