Ordinary differential equation

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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.

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 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 solutions.

Background

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.

Ordinary differential equations (ODEs) arise in many different contexts throughout mathematics and science (social and 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.

Specific mathematical fields include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), geology (weather modelling), chemistry (reaction rates),[1] 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 the object under the force F, which leads to the differential equation

${\displaystyle m{\frac {\mathrm {d} ^{2}x(t)}{\mathrm {d} t^{2}}}=F(x(t)),\,}$

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)).[2][3][4][5]

Definitions

In what follows, let y be a dependent variable and x an 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 Leibniz's notation (dy/dx,d2y/dx2,...dny/dxn) is useful for differentials and when integration is to be done, whereas Newton's and Lagrange's notation (y′,y′′, ... y(n)) is useful for representing derivatives of any order compactly.

General definition of an ODE

Let F be a given function of x, y, and derivatives of y. Then an equation of the form

${\displaystyle F\left(x,y,y',\cdots y^{(n-1)}\right)=y^{(n)}}$

is called an explicit ordinary differential equation of order n.[6][7]

More generally, an implicit ordinary differential equation of order n takes the form:[8]

${\displaystyle F\left(x,y,y',y'',\ \cdots ,\ y^{(n)}\right)=0}$

There are further classifications:

Autonomous

A differential equation not depending on x is called autonomous.

Linear

A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y:

${\displaystyle y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)}$

where ai(x) and r(x) are continuous functions in x.[9][10][11] Non-linear equations cannot be written in this form. The function r(x) is called the source term, leading to two further important classifications:[12][13]

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 yc.

Nonhomogeneous (or inhomogeneous): If r(x) ≠ 0. The additional solution to the complementary function is the particular integral, denoted here by yp.

The general solution to a linear equation can be written as y = yc + yp.

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

${\displaystyle \mathbf {y} ^{(n)}=\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n-1)}\right)}$

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

${\displaystyle {\begin{pmatrix}y_{1}^{(n)}\\y_{2}^{(n)}\\\vdots \\y_{m}^{(n)}\end{pmatrix}}={\begin{pmatrix}F_{1}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n-1)}\right)\\F_{2}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n-1)}\right)\\\vdots \\F_{m}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n-1)}\right)\\\end{pmatrix}}}$

These are not necessarily linear. The implicit analogue is:

${\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n)}\right)={\boldsymbol {0}}}$

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

${\displaystyle {\begin{pmatrix}F_{1}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n)})\\F_{2}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n)})\\\vdots \\F_{m}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\cdots \mathbf {y} ^{(n)})\\\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\\\end{pmatrix}}}$

For a system of the form ${\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}}}$, some sources also require that the Jacobian matrix ${\displaystyle {\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}}$ 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 that (nonsigular) ODE systems.[14][15] Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, Template:Cn although note that any ODE of order greater than one can [and usually is] rewritten as system of ODEs of first order,[16] which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.

Solutions

Given a differential equation

${\displaystyle F\left(x,y,y',\cdots ,y^{(n)}\right)=0}$

a function u: IRR is called the solution or integral curve for F, if u is n-times differentiable on I, and

${\displaystyle F(x,u,u',\ \cdots ,\ u^{(n)})=0\quad x\in I.}$

Given two solutions u: JRR and v: IRR, u is called an extension of v if IJ and

${\displaystyle u(x)=v(x)\quad x\in I.\,}$

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'.[17] A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.[18]

Theories of ODEs

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 did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notable ones being 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 the differential equation meets its limitations very soon unless complex numbers 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 independent variable or variables, and, if so, what are the characteristic properties of this function.

Fuchsian theory

{{#invoke:main|main}} Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed 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, so Clebsch proposed to classify the transcendent functions defined by the 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 more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source, and that ordinary differential equations that admit the same infinitesimal transformations present comparable difficulties of integration. 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.[19]

A general approach to solve DEs 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, Bäcklund transform, and finally finding exact analytic solutions to the DE.

Symmetry methods have been recognized to study differential equations, arising in mathematics, physics, engineering, and many other disciplines.

Sturm–Liouville theory

{{#invoke:main|main}} Sturm–Liouville theory is a theory of eigenvalues and eigenfunctions of linear operators defined in terms of second-order homogeneous linear equations, and is 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

Theorem Assumption Conclusion
Peano existence theorem F continuous local existence only
Picard–Lindelöf theorem F Lipschitz continuous local existence and uniqueness

which are both local results.

Note that 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.[20]

Local existence and uniqueness theorem simplified

The theorem can be stated simply as follows.[21] For the equation and initial value problem:

${\displaystyle y'=F(x,y)\,,\quad y_{0}=y(x_{0})}$

if F and ∂F/∂y are continuous in a closed rectangle

${\displaystyle R=[x_{0}-a,x_{0}+a]\times [y_{0}-b,y_{0}+b]}$

in the x-y plane, where a and b are real (symbolically: a, b ∈ ℝ) and × denotes the cartesian product, square brackets denote closed intervals, then there is an interval

${\displaystyle I=[x_{0}-h,x_{0}+h]\subset [x_{0}-a,x_{0}+a]}$

for some h ∈ ℝ where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F to be linear, this applies to non-linear equations that take the form F(x, y), and it can also be applied to systems of equations.

Global uniqueness and maximum domain of solution

When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:[22]

For each initial condition (x0, y0) there exists a unique maximum (possibly infinite) open interval

${\displaystyle I_{max}=(x_{-},x_{+}),x_{\pm }\in {\mathbb {R} },x_{0}\in I_{max}}$

such that any solution that satisfies this initial condition is a restriction of the solution that satisfies this initial condition with domain Imax.

In the case that ${\displaystyle x_{\pm }\nrightarrow \pm \infty }$, there are exactly two possibilities

where Ω is the open set in which F is defined, and ${\displaystyle \partial {\bar {\Omega }}}$ 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
${\displaystyle y'=y^{2}}$

This means that F(x, y) = y2, which is C1 and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.

Even in such a simple setting, the maximum domain of solution cannot be all ℝ, since the solution is

${\displaystyle y(x)={\frac {y_{0}}{(x_{0}-x)y_{0}+1}}}$

which has maximum domain:

${\displaystyle {\begin{cases}\mathbb {R} &y_{0}=0\\(-\infty ,x_{0}+{\frac {1}{y_{0}}})&y_{0}>0\\(x_{0}+{\frac {1}{y_{0}}},+\infty )&y_{0}<0\end{cases}}}$

This shows clearly that the maximum interval may depend on the initial conditions. The domain of y could be taken as being ${\displaystyle \mathbf {R} \smallsetminus (x_{0}+1/y_{0})}$, 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

${\displaystyle \lim _{x\to x_{\pm }}\|y(x)\|\rightarrow \infty \,,}$

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 differential equation of order n,

${\displaystyle F\left(x,y,y',y'',\ \cdots ,\ y^{(n-1)}\right)=y^{(n)}}$

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

${\displaystyle y_{i}=y^{(i-1)}.\!}$

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

${\displaystyle {\begin{array}{rcl}y_{1}'&=&y_{2}\\y_{2}'&=&y_{3}\\&\vdots &\\y_{n-1}'&=&y_{n}\\y_{n}'&=&F(x,y_{1},\cdots ,y_{n}).\end{array}}}$

more compactly in vector notation:

${\displaystyle \mathbf {y} '=\mathbf {F} (x,\mathbf {y} )}$

where

${\displaystyle \mathbf {y} =(y_{1},\cdots ,y_{n}),\quad \mathbf {F} (x,y_{1},\cdots ,y_{n})=(y_{2},\cdots ,y_{n},F(x,y_{1},\cdots ,y_{n})).}$

Summary of exact solutions

Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.

In the table below, P(x), Q(x), P(y), Q(y), and M(x,y), N(x,y) are any integrable functions of x, y, and b and c are real given constants, and C1, C2,... are arbitrary constants (complex in general). The differential equations are in their equivalent and alternative forms that lead to the solution through integration.

In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in summation), and the notation ∫xF(λ)dλ just means to integrate F(λ) with respect to λ, then after the integration substitute λ = x, without adding constants (explicitly stated).

Differential equation Solution method General solution
Separable equations
First-order, separable in x and y (general case, see below for special cases)[23] Separation of variables (divide by P2Q1). ${\displaystyle \int ^{x}{\frac {P_{1}(\lambda )}{P_{2}(\lambda )}}\,d\lambda +\int ^{y}{\frac {Q_{2}(\lambda )}{Q_{1}(\lambda )}}\,d\lambda =C\,\!}$
First-order, separable in x[21] Direct integration. ${\displaystyle y=\int ^{x}F(\lambda )\,d\lambda +C\,\!}$
First-order, autonomous, separable in y[21] Separation of variables (divide by F). ${\displaystyle x=\int ^{y}{\frac {d\lambda }{F(\lambda )}}+C\,\!}$
First-order, separable in x and y[21] Integrate throughout. ${\displaystyle \int ^{y}P(\lambda )\,{d\lambda }+\int ^{x}Q(\lambda )\,d\lambda =C\,\!}$
General first-order equations
First-order, homogeneous[21] Set y = ux, then solve by separation of variables in u and x. ${\displaystyle \ln(Cx)=\int ^{y/x}{\frac {d\lambda }{F(\lambda )-\lambda }}\,\!}$
First-order, separable[23] Separation of variables (divide by xy).

If N = M, the solution is xy = C.

Exact differential, first-order[21] Integrate throughout. {\displaystyle {\begin{aligned}F(x,y)&=\int ^{y}M(x,\lambda )\,d\lambda +\int ^{x}N(\lambda ,y)\,d\lambda \\&+Y(y)+X(x)=C\end{aligned}}\,\!}

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, first-order[21] Integration factor μ(x, y) satisfying If μ(x, y) can be found:
General second-order equations
Second-order, autonomous[24] Multiply equation by 2dy/dx, substitute ${\displaystyle 2{\frac {dy}{dx}}{\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right)^{2}\,\!}$, then integrate twice. ${\displaystyle x=\pm \int ^{y}{\frac {d\lambda }{\sqrt {2\int ^{\lambda }F(\epsilon )\,d\epsilon +C_{1}}}}+C_{2}\,\!}$
Linear equations (up to nth order)
First-order, linear, inhomogeneous, function coefficients[21] Integrating factor: ${\displaystyle e^{\int ^{x}P(\lambda )\,d\lambda }}$. ${\displaystyle y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left[\int ^{x}e^{\int ^{\lambda }P(\epsilon )\,d\epsilon }Q(\lambda )\,{d\lambda }+C\right]}$
Second-order, linear, inhomogeneous, constant coefficients[25] Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions ${\displaystyle e^{\alpha _{j}x}}$.

Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.[21]

${\displaystyle y=y_{c}+y_{p}}$

If b2 > 4c, then:

If b2 = 4c, then:

If b2 < 4c, then:

nth-order, linear, inhomogeneous, constant coefficients[25] Complementary function yc: assume yc = eαx, substitute and solve polynomial in α, to find the linearly independent functions ${\displaystyle e^{\alpha _{j}x}}$.

Particular integral yp: in general the method of variation of parameters, though for very simple r(x) inspection may work.[21]

${\displaystyle y=y_{c}+y_{p}}$

Since αj are the solutions of the polynomial of degree n: ${\displaystyle \prod _{j=1}^{n}\left(\alpha -\alpha _{j}\right)=0\,\!}$, then:

for αj all different,

for each root αj repeated kj times,

for some αj complex, then setting α = χj + iγj, and using Euler's formula, allows some terms in the previous results to be written in the form

${\displaystyle C_{j}e^{\alpha _{j}x}=C_{j}e^{\chi _{j}x}\cos(\gamma _{j}x+\phi _{j})\,\!}$

where ϕj is an arbitrary constant (phase shift).

Notes

1. Mathematics for Chemists, D.M. Hirst, Macmillan Press, 1976, (No ISBN) SBN: 333-18172-7
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21. 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
22. Boscain; Chitour 2011, p. 21
23. 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
24. Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978, ISBN 0-7135-1594-5
25. 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

References

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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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• 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.