Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable, and, less commonly, in contrast with stochastic differential equations (SDEs) where the modeled process is random.

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 \(a_0(x),\ldots,a_n(x)\) and \(b(x)\) are functions that do not need to be linear, and \(y',\ldots, y^{(n)}\) are the successive derivatives of the unknown function \(y\) of the variable \(x\).

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.

Ordinary 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 modeling), chemistry (reaction rates), biology (infectious diseases, genetic variation), ecology and population modeling (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 \(F(x(t))\).

In what follows, \(y\) is a dependent variable representing an unknown function \(y=f(x)\) of the independent variable \(x\). 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 \(\frac{dy}{dx},\frac{d^2y}{dx^2},\ldots, \frac{d^ny}{dx^n}\) is more useful for differentiation and integration, whereas Lagrange's notation \(y',y'',\ldots,y^{(n)}\) is more useful for representing higher-order derivatives 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.

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

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

is called an explicit ordinary differential equation of order \(n\).

More generally, an implicit ordinary differential equation of order \(n\) takes the form:

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

There are further classifications:

Autonomous

A differential equation is autonomous if it does not depend on the variable x.

Linear

A differential equation is linear if \(F\) can be written as a linear combination of the derivatives of \(y\); that is, it can be rewritten as

\(y^{(n)} = \sum_{i=0}^{n - 1} a_i(x) y^{(i)} + r(x)\)

where \(a_i(x)\) and \(r(x)\) are continuous functions of \(x\).

Non-linear

A differential equation that is not linear.

A number of coupled differential equations form a system of equations. If \(\mathbf{y}\) is a vector whose elements are functions; \(\mathbf{y}(x) = [y_1(x),y_2(x),\ldots, y_m(x)]\), and \(\mathbf{F}\) is a vector-valued function of \(\mathbf{y}\) 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 \(\boldsymbol{0}=(0,0,\ldots,0)\) is the zero vector. In matrix form

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

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 \(\frac{\partial\mathbf{F}(x,\mathbf{u},\mathbf{v})}{\partial \mathbf{v}}\) 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 than (nonsingular) ODE systems. Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.

The behavior of a system of ODEs can be visualized through the use of a phase portrait.

Given a differential equation

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

a function \(u:I\subset\mathbb{R}\to\mathbb{R}\), where \(I\) is an interval, 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 \(u:J\subset\mathbb{R}\to\mathbb{R}\) and \(v:I\subset\mathbb{R}\to\mathbb{R}\), \(u\) is called an extension of \(v\) if \(I\subset J\) and

\(u(x) = v(x) \quad x \in I.\,\)

A solution that has no extension is called a maximal solution. A solution defined on all of \(\mathbb{R}\) is called a global solution.

A general solution of an \(n\)th-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'. A singular solution is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.

In the context of linear ODE, the terminology particular solution can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the homogeneous solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the guessing method section in this article, and is frequently used when discussing the method of undetermined coefficients and variation of parameters.

For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on the whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of Lipschitz differential equations.

As example, the equation

\(y'= -\text{sgn}(y)\sqrt{|y|},\,\,y(0)=1\)

admits the finite duration solution:

\(y(x)=\frac{1}{4}\left(1-\frac{x}{2}+\left|1-\frac{x}{2}\right|\right)^2\)

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.

The primitive attempt in dealing with differential equations had in view a reduction to quadratures, that is, expressing the solutions in terms of known function and their integrals. This is possible for linear equations with constant coefficients, it appeared in the 19th century that this is generally impossible in other cases. Hence, analysts began the study (for their own) of functions that are solutions of differential equations, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by quadratures, but whether a given differential equation suffices for the definition of a function, and, if so, what are the characteristic properties of such functions.

Two memoirs by Fuchs inspired a novel approach, subsequently elaborated by Thomé and Frobenius. (WHO?)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.

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.

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 non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform, and finally finding exact analytic solutions to DE.

Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.

Sturm-Liouville theory 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. SLPs are also useful in the analysis of certain partial differential equations.

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

In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of Grönwall's inequality are met.

Also, uniqueness theorems like the Lipschitz one above do not apply to DAE systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone.