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| {{Differential equations}}
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| {{see also|Examples of differential equations}}
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| In [[mathematics]], an '''ordinary differential equation''' or '''ODE''' is an equation containing a function of one [[independent variable]] and its derivatives. The term "''ordinary''" is used in contrast with the term [[partial differential equation]] which may be with respect to ''more than'' one independent variable.
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| Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by [[elementary functions]] in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and [[numerical ordinary differential equations|numerical]] methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solution.
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| ==Background==
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| [[Image:Parabolic trajectory.svg|right|thumb|250px|The [[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.]]
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| Ordinary differential equations (ODEs) arise in many different contexts throughout mathematics and science ([[social science|social]] and [[natural science|natural]]) one way or another, because when describing changes mathematically, the most accurate way uses differentials and derivatives (related, though not quite the same). Since various differentials, derivatives, and functions become inevitably related to each other via equations, a differential equation is the result, describing dynamical phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (time derivatives), or gradients of quantities, which is how they enter differential equations.
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| Specific mathematical fields include [[geometry]] and [[analytical mechanics]]. Scientific fields include much of [[physics]] and [[astronomy]] (celestial mechanics), [[geology]] (weather modelling), [[chemistry]] (reaction rates),<ref>Mathematics for Chemists, D.M. Hirst, [[Macmillan Publishers|Macmillan Press]], 1976, (No ISBN) SBN: 333-18172-7</ref> [[biology]] (infectious diseases, genetic variation), [[ecology]] and [[population modelling]] (population competition), [[economics]] (stock trends, interest rates and the market equilibrium price changes).
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| Many mathematicians have studied differential equations and contributed to the field, including [[Isaac Newton|Newton]], [[Gottfried Leibniz|Leibniz]], the [[Bernoulli family]], [[Riccati]], [[Alexis Claude Clairaut|Clairaut]], [[d'Alembert]], and [[Euler]].
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| A simple example is [[Newton's second law]] of motion — the relationship between the displacement ''x'' and the time ''t'' of the object under the force ''F'', which leads to the differential equation | |
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| :<math>m \frac{\mathrm{d}^2 x(t)}{\mathrm{d}t^2} = F(x(t)),\,</math>
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| for the motion of a particle of constant mass ''m''. In general, ''F'' depends on the position ''x''(''t'') of the particle at time ''t'', and so the unknown function ''x''(''t'') appears on both sides of the differential equation, as is indicated in the notation ''F''(''x''(''t'')).<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>
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| ==Definitions==
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| In what follows, let ''y'' be a [[Dependent and independent variables|dependent variable]] and ''x'' an [[Dependent and independent variables|independent variable]], so that ''y'' = ''y''(''x'') is an unknown function in ''x''. The [[notation for differentiation]] varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the [[notation for differentiation#Leibniz's notation|Leibniz's notation]] (''dy/dx'',''d''<sup>2</sup>''y/dx''<sup>2</sup>,...''d<sup>n</sup>y/dx<sup>n</sup>'') is useful for ''differentials'' and when [[Integration (mathematics)|integration]] is to be done, whereas [[notation for differentiation#Newton's notation|Newton's]] and [[notation for differentiation#Lagrange's notation|Lagrange's notation]] (''y′'',''y′′'', ... ''y''<sup>(''n'')</sup>) is useful for representing ''derivatives'' of any order compactly.
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| ===General definition of an ODE===
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| Let ''F'' be a given function of ''x'', ''y'', and derivatives of ''y''. Then an equation of the form
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| :<math>F\left (x,y,y',\cdots y^{(n-1)} \right )=y^{(n)}</math>
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| is called an [[Implicit and explicit functions|explicit]] ''ordinary differential equation'' of ''order'' ''n''.<ref>{{harvtxt|Harper|1976|p=127}}</ref><ref>{{harvtxt|Kreyszig|1972|p=2}}</ref>
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| More generally, an ''[[Implicit and explicit functions|implicit]]'' ordinary differential equation of order ''n'' takes the form:<ref>{{harvtxt|Simmons|1972|p=3}}</ref>
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| :<math>F\left(x, y, y', y'',\ \cdots,\ y^{(n)}\right) = 0</math>
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| There are further classifications:
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| ;[[Autonomous system (mathematics)|Autonomous]]
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| A differential equation not depending on ''x'' is called ''[[Autonomous system (mathematics)|autonomous]]''.
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| ;[[Linear differential equation|Linear]]
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| A differential equation is said to be ''linear'' if ''F'' can be written as a [[linear combination]] of the derivatives of ''y'':
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| :<math>y^{(n)} = \sum_{i=0}^{n-1} a_i(x) y^{(i)} + r(x)</math>
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| where ''a''<sub>''i''</sub>(''x'') and ''r''(''x'') continuous functions in ''x''.<ref>{{harvtxt|Harper|1976|p=127}}</ref><ref>{{harvtxt|Kreyszig|1972|p=24}}</ref><ref>{{harvtxt|Simmons|1972|p=47}}</ref> '''[[Non-linear differential equation|Non-linear]]''' equations cannot be written in this form. The function ''r''(''x'') is called the ''source term'', leading to two further important classifications:<ref>{{harvtxt|Harper|1976|p=128}}</ref><ref>{{harvtxt|Kreyszig|1972|p=24}}</ref>
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| '''Homogeneous:''' If ''r''(''x'') = 0, and consequently one "automatic" solution is the [[trivial solution]], ''y'' = 0. The solution of a linear homogeneous equation is a '''complementary function''', denoted here by ''y<sub>c</sub>''.
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| '''Nonhomogeneous (or inhomogeneous):''' If ''r''(''x'') ≠ 0. The additional solution to the complementary function is the '''particular integral''', denoted here by ''y<sub>p</sub>''.
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| The general solution to a linear equation can be written as ''y'' = ''y<sub>c</sub>'' + ''y<sub>p</sub>''.
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| ===System of ODEs===
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| 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]] of '''y''' and its derivatives, then
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| :<math>\mathbf{y}^{(n)} = \mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right)</math>
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| is an ''explicit system of ordinary differential equations'' of ''order or dimension'' ''m''. In [[column vector]] form:
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| :<math>\begin{pmatrix}
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| y_1^{(n)} \\
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| y_2^{(n)} \\
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| \vdots \\
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| y_m^{(n)}
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| \end{pmatrix} =
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| \begin{pmatrix}
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| F_1 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right ) \\
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| F_2 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right ) \\
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| \vdots \\
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| F_m \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right) \\
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| \end{pmatrix}</math>
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| These are not necessarily linear. The ''implicit'' analogue is:
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| :<math>\mathbf{F} \left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)} \right) = \boldsymbol{0}</math>
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| where '''0''' = (0, 0,... 0) is the [[zero vector]]. In matrix form
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| :<math>\begin{pmatrix}
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| F_1(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\
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| F_2(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\
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| \vdots \\
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| F_m(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\
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| \end{pmatrix}=\begin{pmatrix}
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| 0\\
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| 0\\
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| \vdots\\
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| 0\\
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| \end{pmatrix}</math>
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| === Solutions ===
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| Given a differential equation
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| :<math>F\left(x, y, y', \cdots, y^{(n)} \right) = 0</math>
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| a function {{nowrap|''u'': ''I'' ⊂ '''R''' → '''R'''}} is called the ''solution'' or [[integral curve]] for ''F'', if ''u'' is ''n''-times differentiable on ''I'', and
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| :<math>F(x,u,u',\ \cdots,\ u^{(n)})=0 \quad x \in I.</math>
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| 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
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| :<math>u(x) = v(x) \quad x \in I.\,</math>
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| A solution that has no extension is called a ''maximal solution''. A solution defined on all of '''R''' is called a ''global solution''.
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| 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]] 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>
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| ==Theories of ODEs==
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| ===Singular solutions===
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| The theory of [[singular solution]]s of ordinary and [[partial differential equation]]s was a subject of research from the time
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| of Leibniz, but only since the middle of the nineteenth century did it
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| receive special attention. A valuable but little-known work on the
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| subject is that of Houtain (1854). [[Jean Gaston Darboux|Darboux]] (starting in 1873) was a
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| leader in the theory, and in the geometric interpretation of these
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| solutions he opened a field worked by various
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| writers, notable ones being [[Felice Casorati (mathematician)|Casorati]] and [[Arthur Cayley|Cayley]]. To the latter is due (1872)
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| the theory of singular solutions of differential equations of the
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| first order as accepted circa 1900.
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| ===Reduction to quadratures===
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| 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. [[Carl Friedrich Gauss|Gauss]] (1799) showed, however, that the differential equation meets its limitations very soon unless [[complex number]]s are introduced. Hence, analysts began to substitute the study of functions, thus opening a new and fertile field. [[Cauchy]] was the first to appreciate the importance of this view. Thereafter, the real question was to be not whether a solution is possible by means of known functions or their integrals but whether a given differential equation suffices for the definition of a function of the
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| independent variable or variables, and, if so, what are the characteristic properties of this function.
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| ===Fuchsian theory===
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| {{main|Frobenius method}}
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| Two memoirs by [[Lazarus Fuchs|Fuchs]] (''Crelle'', 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and [[Ferdinand Georg Frobenius|Frobenius]]. Collet was a prominent contributor beginning in 1869, although his method for integrating a
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| non-linear system was communicated to Bertrand in 1868. [[Alfred Clebsch|Clebsch]] (1873) attacked
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| the theory along lines parallel to those followed in his theory of
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| [[Abelian integral]]s. As the latter can be classified according to the
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| properties of the fundamental curve that remains unchanged under a
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| rational transformation, so Clebsch proposed to classify the
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| transcendent functions defined by the differential equations
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| according to the invariant properties of the corresponding surfaces
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| ''f'' = 0 under rational one-to-one transformations.
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| ===Lie's theory===
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| From 1870, [[Sophus Lie]]'s work put the theory of differential equations
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| on a more satisfactory foundation. He showed that the integration
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| theories of the older mathematicians can, by the introduction of what are now called [[Lie group]]s, be referred to a common source, and that ordinary differential equations that admit the same [[infinitesimal transformation]]s present comparable difficulties of integration. He also emphasized the subject of [[contact transformation|transformations of contact]].
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| 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>
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| A general approach to solve DEs uses the symmetry property of differential equations, the continuous [[infinitesimal transformation]]s 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 pair]]s, recursion operators, [[Bäcklund transform]], and finally finding exact analytic solutions to the DE.
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| Symmetry methods have been recognized to study differential equations, arising in mathematics, physics, engineering, and many other disciplines.
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| === Sturm–Liouville theory ===
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| {{main|Sturm–Liouville theory}}
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| Sturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linear
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| operators defined in terms of second-order homogeneous linear equations, and is useful
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| in the analysis of certain partial differential equations.
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| ==Existence and uniqueness of solutions==
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| 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
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| :{| class="wikitable"
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| |-
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| ! Theorem
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| ! Assumption
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| ! Conclusion
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| |-
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| |[[Peano existence theorem]]
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| ||F [[continuous function|continuous]]
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| ||local existence only
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| |-
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| |[[Picard–Lindelöf theorem]]
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| ||F [[Lipschitz continuous]]
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| ||local existence and uniqueness
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| |-
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| |}
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| which are both local results.
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| ===Local existence and uniqueness theorem simplified===
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| 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:
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| :<math> y' = F(x,y)\,,\quad y_0 = y(x_0)</math>
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| if ''F'' and ∂''F''/∂''y'' are continuous in a closed rectangle
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| :<math>R=[x_0-a,x_0+a]\times [y_0-b,y_0+b]</math>
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| in the ''x-y'' plane, where ''a'' and ''b'' are [[real number|real]] (symbolically: ''a, b'' ∈ ℝ) and × denotes the [[cartesian product]], square brackets denote [[interval notation|closed intervals]], then there is an interval
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| :<math>I = [x_0-h,x_0+h] \subset [x_0-a,x_0+a]</math>
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| for some ''h'' ∈ ℝ where ''the'' solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on ''F'' to be linear, this applies to non-linear equations that take the form ''F''(''x, y''), and it can also be applied to systems of equations.
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| ===Global uniqueness and maximum domain of solution===
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| 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. 21</ref>
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| For each initial condition (''x''<sub>0</sub>, ''y''<sub>0</sub>) there exists a unique maximum (possibly infinite) open interval
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| :<math>I_{max} = (x_-,x_+), x_\pm \in \mathbb{R}, x_0 \in I_{max}</math>
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| such that any solution that satisfies this initial condition is a [[Restriction (mathematics)|restriction]] of the solution that satisfies this initial condition with domain ''I''<sub>max</sub>.
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| In the case that <math>x_\pm \nrightarrow \pm\infty</math>, there are exactly two possibilities
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| *explosion in finite time: <math>\lim_{x \to x_\pm} \|y(x)\| \rightarrow \infty</math>
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| *leaves domain of definition: <math>\lim_{x \to x_\pm} \in \partial \bar{\Omega}</math>
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| where Ω is the open set in which ''F'' is defined, and <math>\partial \bar{\Omega}</math> is its boundary.
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| Note that the maximum domain of the solution
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| * is always an interval (to have uniqueness)
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| * may be smaller than ℝ
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| * may depend on the specific choice of (''x''<sub>0</sub>, ''y''<sub>0</sub>).
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| ;Example
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| :<math>y' = y^2</math>
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| This means that ''F''(''x, y'') = ''y''<sup>2</sup>, which is ''C''<sup>1</sup> and therefore Lipschitz continuous for all ''y'', satisfying the Picard–Lindelöf theorem.
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| Even in such a simple setting, the maximum domain of solution cannot be all ℝ, since the solution is
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| :<math>y(x) = \frac{y_0}{(x_0-x)y_0+1}</math>
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| which has maximum domain:
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| :<math>\begin{cases} \mathbb{R} & y_0 = 0 \\
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| (-\infty, x_0+\frac{1}{y_0}) & y_0 > 0 \\
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| (x_0+\frac{1}{y_0},+\infty) & y_0 < 0 \end{cases}</math>
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| This shows clearly that the maximum interval may depend on the initial conditions. The domain of ''y'' could be taken as being <math>\mathbf{R} \smallsetminus (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.
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| The maximum domain is not ℝ because
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| :<math>\lim_{x \to x_\pm} \|y(x)\| \rightarrow \infty\,,</math>
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| which is one of the two possible cases according to the above theorem.
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| == Reduction of order ==
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| Differential equations can usually be solved more easily if the order of the equation can be reduced.
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| === Reduction to a first-order system === <!-- Redirect [[Stiff equation]] links here -->
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| Any differential equation of order ''n'',
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| :<math>F\left(x, y, y', y'',\ \cdots,\ y^{(n-1)}\right) = y^{(n)}</math>
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| can be written as a system of ''n'' first-order differential equations by defining a new family of unknown functions
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| :<math>y_i = y^{(i-1)}.\!</math>
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| for ''i'' = 1, 2,... ''n''. The ''n''-dimensional system of first-order coupled differential equations is then
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| :<math>\begin{array}{rcl}
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| y_1'&=&y_2\\
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| y_2'&=&y_3\\
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| &\vdots&\\
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| y_{n-1}'&=&y_n\\
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| y_n'&=&F(x,y_1,\cdots,y_n).
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| \end{array}
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| </math>
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| more compactly in vector notation:
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| :<math>\mathbf{y}'=\mathbf{F}(x,\mathbf{y})</math>
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| where
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| :<math>\mathbf{y}=(y_1,\cdots,y_n),\quad \mathbf{F}(x,y_1,\cdots,y_n)=(y_2,\cdots,y_n,F(x,y_1,\cdots,y_n)).</math>
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| ==Summary of exact solutions==
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| <!--Please leave the table alone. If a table format is really disliked then instead of simply deleting it - expand into prose. For now it is to summarize some common forms of equations and their solutions. Details are in the other articles. -->
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| Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.
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| 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 ''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.
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| In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in [[summation]]), and the notation ∫<sup>''x''</sup>''F''(λ)dλ just means to integrate ''F''(λ) with respect to λ, then ''after'' the integration substitute λ = ''x'', without adding constants (explicitly stated).
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| :{| class="wikitable"
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| ! scope="col" width="200px" | Differential equation
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| ! scope="col" width="100px" | Solution method
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| ! scope="col" width="400px" | General solution
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| ! colspan="3" | Separable equations
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| | 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, Schuam's Outline Series, 2009, ISC_2N 978-0-07-154855-7</ref>
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| <math> P_1(x)Q_1(y) + P_2(x)Q_2(y)\,\frac{dy}{dx} = 0 \,\!</math>
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| <math> P_1(x)Q_1(y)\,dx + P_2(x)Q_2(y)\,dy = 0 \,\!</math>
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| || Separation of variables (divide by ''P''<sub>2</sub>''Q''<sub>1</sub>).
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| || <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>
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| | First-order, separable in ''x''<ref name= "EDEBVP" />
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| <math>\frac{dy}{dx} = F(x)\,\!</math>
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| <math>dy= F(x) \, dx\,\!</math>
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| ||Direct integration.
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| || <math>y= \int^x F(\lambda) \, d\lambda + C \,\!</math>
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| | First-order, autonomous, separable in ''y''<ref name= "EDEBVP" />
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| <math>\frac{dy}{dx} = F(y)\,\!</math>
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| <math>dy= F(y) \, dx\,\!</math>
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| || [[Separation of variables]] (divide by ''F'').
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| || <math>x=\int^y \frac{d\lambda}{F(\lambda)}+C\,\!</math>
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| | First-order, separable in ''x'' and ''y''<ref name= "EDEBVP" />
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| <math>P(y)\frac{dy}{dx} + Q(x)= 0\,\!</math>
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| <math>P(y)\,dy + Q(x)\,dx =0\,\!</math>
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| || Integrate throughout.
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| || <math>\int^y P(\lambda)\,{d\lambda} + \int^x Q(\lambda)\,d\lambda = C\,\!</math>
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| ! colspan="3"| General first-order equations
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| | First-order, homogeneous<ref name= "EDEBVP" />
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| <math>\frac{dy}{dx} = F \left( \frac{y}{x} \right ) \,\!</math>
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| || Set ''y = ux'', then solve by separation of variables in ''u'' and ''x''.
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| || <math> \ln (Cx) = \int^{y/x} \frac{d\lambda}{F(\lambda) - \lambda} \, \! </math>
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| | First-order, separable<ref name= "MHFT" />
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| <math> yM(xy) + xN(xy)\,\frac{dy}{dx} = 0 \,\!</math>
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| <math> yM(xy)\,dx + xN(xy)\,dy = 0 \,\!</math>
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| || Separation of variables (divide by ''xy'').
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| ||
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| <math> \ln (Cx) = \int^{xy} \frac{N(\lambda)\,d\lambda}{\lambda [N(\lambda)-M(\lambda)] } \,\!</math>
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| If ''N'' = ''M'', the solution is ''xy = C''.
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| | [[Exact differential]], first-order<ref name= "EDEBVP" />
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| <math> M(x,y) \frac{dy}{dx} + N(x,y) = 0 \,\!</math>
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| <math> M(x,y)\,dy + N(x,y)\,dx = 0 \,\!</math>
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| where <math> \frac{\partial M}{\partial x} = \frac{\partial N}{\partial y} \, \! </math>
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| || Integrate throughout.
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| || <math> \begin{align}
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| F(x,y) & = \int^y M(x,\lambda)\,d\lambda + \int^x N(\lambda,y)\,d\lambda \\
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| & + Y(y) + X(x) = C
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| \end{align} \,\!</math>
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| 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.
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| | [[Inexact differential]], first-order<ref name= "EDEBVP" />
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| <math> M(x,y) \frac{dy}{dx} + N(x,y) = 0 \,\!</math>
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| <math> M(x,y)\,dy + N(x,y)\,dx = 0 \,\!</math>
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| where <math> \frac{\partial M}{\partial x} \neq \frac{\partial N}{\partial y} \, \! </math>
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| || [[Integration factor]] μ(''x, y'') satisfying
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| <math> \frac{\partial (\mu M)}{\partial x} = \frac{\partial (\mu N)}{\partial y} \, \! </math>
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| || If μ(''x, y'') can be found:
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| <math> \begin{align}
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| F(x,y) & = \int^y \mu(x,\lambda)M(x,\lambda)\,d\lambda + \int^x \mu(\lambda,y)N(\lambda,y)\,d\lambda \\
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| & + Y(y) + X(x) = C \\
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| \end{align} \, \! </math>
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| ! colspan="3"| General second-order equations
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| | Second-order, autonomous<ref>Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978, ISBN 0-7135-1594-5</ref>
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| <math>\frac{d^2y}{dx^2} = F(y) \,\!</math>
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| || Multiply equation by 2''dy/dx'', substitute <math>2 \frac{dy}{dx}\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)^2 \,\!</math>, then integrate twice.
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| || <math> x = \pm \int^y \frac{ d \lambda}{\sqrt{2 \int^\lambda F(\epsilon) \, d \epsilon + C_1}} + C_2 \, \! </math>
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| ! colspan="3"| Linear equations (up to ''n''th order)
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| | First-order, linear, inhomogeneous, function coefficients<ref name= "EDEBVP" />
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| <math>\frac{dy}{dx} + P(x)y=Q(x)\,\!</math>
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| || Integrating factor: <math>e^{\int^x P(\lambda)\,d\lambda}</math>.
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| || <math>y = e^{- \int^x P(\lambda) \, d\lambda}\left[\int^x e^{\int^\lambda P(\epsilon) \, d\epsilon}Q(\lambda) \, {d\lambda} +C \right]</math>
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| | 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>
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| <math>\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = r(x)\,\!</math>
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| || Complementary function ''y<sub>c</sub>'': assume ''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>.
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| 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" />
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| || <math>y=y_c+y_p</math>
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| If ''b''<sup>2</sup> > 4''c'', then:
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| <math>y_c=C_1e^{ \left ( -b+\sqrt{b^2 - 4c} \right )\frac{x}{2}} + C_2e^{-\left ( b+\sqrt{b^2 - 4c} \right )\frac{x}{2}}\,\!</math>
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| If ''b''<sup>2</sup> = 4''c'', then:
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| <math>y_c = (C_1x + C_2)e^{-bx/2}\,\!</math>
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| If ''b''<sup>2</sup> < 4''c'', then:
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| <math> y_c = e^{ -b\frac{x}{2}} \left [ C_1 \sin{\left ( \sqrt{\left | b^2-4c \right |}\frac{x}{2} \right )} + C_2\cos{\left ( \sqrt{\left | b^2-4c \right |}\frac{x}{2} \right )} \right ] \,\!</math>
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| | ''n''th-order, linear, inhomogeneous, constant coefficients<ref name= "MMPE" />
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| <math> \sum_{j=0}^n b_j \frac{d^j y}{dx^j} = r(x)\,\!</math>
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| || Complementary function ''y<sub>c</sub>'': assume ''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>.
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| 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" />
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| || <math>y=y_c+y_p</math>
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| Since α<sub>''j''</sub> are the solutions of the [[polynomial]] of [[Degree of a polynomial|degree]] ''n'': <math> \prod_{j=1}^n \left ( \alpha - \alpha_j \right ) = 0 \,\!</math>, then:
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| for α<sub>''j''</sub> all different,
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| <math> y_c = \sum_{j=1}^n C_j e^{\alpha_j x} \,\!</math>
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| for each root α<sub>''j''</sub> repeated ''k<sub>j</sub>'' times,
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| <math> y_c = \sum_{j=1}^n \left( \sum_{\ell=1}^{k_j} C_\ell x^{\ell-1}\right )e^{\alpha_j x} \,\!</math>
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| for some α<sub>''j''</sub> complex, then setting α = χ<sub>''j''</sub> + ''i''γ<sub>''j''</sub>, and using [[Euler's formula]], allows some terms in the previous results to be written in the form
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| :<math> C_je^{\alpha_j x} = C_j e^{\chi_j x}\cos(\gamma_j x + \phi_j)\,\!</math>
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| where ϕ<sub>''j''</sub> is an arbitrary constant (phase shift).
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| |}
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| == Software for ODE solving ==
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| * [[FuncDesigner]] ([[BSD licenses|BSD-licensed]], uses [[Automatic differentiation]])
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| * [[Maxima (software)|Maxima]] [[computer algebra system]] ([[GNU General Public License|GPL]])
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| * [[COPASI]] a free ([[Artistic license|Artistic License 2.0]]) software package for the integration and analysis of ODEs.
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| * [[MATLAB]] a Technical Computing Software (MATrix LABoratory)
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| * [[GNU Octave]] a high-level language, primarily intended for numerical computations.
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| * [[Scilab]] an open source software for numerical computation.
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| * [[Maple (software)|Maple]]
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| * [[Mathematica]]
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| * [[Julia (programming language)]]
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| * [[SciPy]] a Python package that includes an ODE integration module.
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| ==See also==
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| *[[Boundary value problem]]
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| *[[Difference equation]]
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| *[[Laplace transform applied to differential equations]]
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| *[[List of dynamical systems and differential equations topics]]
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| *[[Matrix differential equation]]
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| *[[Method of undetermined coefficients]]
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| *[[Numerical ordinary differential equations]]
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| *[[Separation of variables]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| <references/>
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| * {{citation | first1 = David | last1 = Halliday | first2 = Robert | last2 = Resnick | year = 1977 | isbn = 0-471-71716-9 | title = Physics | edition = 3rd | publisher = [[John Wiley & Sons|Wiley]] | location = New York }}
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| * {{citation | first1 = Charlie | last1 = Harper | year = 1976 | isbn = 0-13-487538-9 | title = Introduction to Mathematical Physics | publisher = [[Prentice-Hall]] | location = New Jersey }}
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| * {{citation | first1 = Erwin | last1 = Kreyszig | year = 1972 | isbn = 0-471-50728-8 | title = Advanced Engineering Mathematics | edition = 3rd | publisher = [[John Wiley & Sons|Wiley]] | location = New York }}.
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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
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| * {{citation | first1 = George F. | last1 = Simmons | year = 1972 | title = Differential Equations with Applications and Historical Notes | publisher = [[McGraw-Hill]] | location = New York | lccn = 75-173716 }}
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| * {{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 }}
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| * {{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 = french}}
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| * {{citation | first1 = Dresner | last1 = Lawrence | year = 1999 | title = Applications of Lie's Theory of Ordinary and Partial Differential Equations | publisher = [[Institute of Physics Publishing]] | location = Bristol and Philadelphia }}
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| == Bibliography ==
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| * {{cite book | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | publisher=[[McGraw-Hill]] | location=New York | year=1955}}
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| * Hartman, Philip, ''Ordinary Differential Equations, 2nd Ed.'', Society for Industrial & Applied Math, 2002. ISBN 0-89871-510-5.
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| * 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]
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| * E. L. Ince, ''Ordinary Differential Equations'', Dover Publications, 1958, ISBN 0-486-60349-0
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| * [[Witold Hurewicz]], ''Lectures on Ordinary Differential Equations'', Dover Publications, ISBN 0-486-49510-8
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| *{{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 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}.
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| * {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
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| * 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
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| * D. Zwillinger, ''Handbook of Differential Equations (3rd edition)'', Academic Press, Boston, 1997.
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| ==External links==
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| * {{springer|title=Differential equation, ordinary|id=p/d031910}}
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| {{wikibooks|Calculus/Ordinary differential equations}}
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| *{{dmoz|Science/Math/Differential_Equations/|Differential Equations}} (includes a list of software for solving differential equations).
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| *[http://eqworld.ipmnet.ru/index.htm EqWorld: The World of Mathematical Equations], containing a list of ordinary differential equations with their solutions.
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| *[http://tutorial.math.lamar.edu/classes/de/de.aspx Online Notes / Differential Equations] by Paul Dawkins, [[Lamar University]].
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| *[http://www.sosmath.com/diffeq/diffeq.html Differential Equations], S.O.S. Mathematics.
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| *[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.
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| * [http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ Ordinary Differential Equations and Dynamical Systems] lecture notes by [[Gerald Teschl]].
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| * [http://www.jirka.org/diffyqs/ Notes on Diffy Qs: Differential Equations for Engineers] An introductory textbook on differential equations by Jiri Lebl of [[UIUC]].
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| * [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.
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| [[Category:Differential calculus]]
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| [[Category:Ordinary differential equations|*]]
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