Differential equation

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Differential equation
Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation.

Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions —the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Directions of study

The study of differential equations is a wide field in pure and applied mathematics, physics, meteorology, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist, and it also results in more physically reasonable properties of solutions, such as possible presence of shocks for equations of hyperbolic type.

The study of the stability of solutions of differential equations is known as stability theory.

Nomenclature

The theory of differential equations is quite developed and the methods used to study them vary significantly with the type of the equation.

• An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.

• Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation
$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0$
(in which y is the dependent variable) is a second-order differential equation. In the classical literature also distinction is made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form.
• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

Both ordinary and partial differential equations are broadly classified as linear and nonlinear. A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.

There are very few methods of explicitly solving nonlinear differential equations; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Classification summary

The mathematical definitions for the various classifications of differential equation can be collected as follows.

Ordinary DE classification

See main article: Ordinary differential equation.

In the table below,

All differential equations are of order n and arbitrary degree d.

F is an implicit function of: an independent variable x, a dependent variable y (a function of x), and integer derivatives of y (fractional derivatives are in fact possible, but not considered here).

y may in general be a vector valued function:

$y = \begin{pmatrix} y_1 (x) \\ y_2 (x) \\ \vdots \\ y_m (x), \end{pmatrix} \,\!$

so x is an element of the vector space R, y an element of a vector space of dimension m, $x \in \mathbf{R}, y \in \mathbf{R}^m \,\!$, where R is the set of real numbers, $\mathbf{R}^m = \mathbf{R} \times \mathbf{R} \times \cdots \times \mathbf{R} \,\!$ is the cartesian product of R with itself m times to form an m-tuple of real numbers.

This leads to a system of differential equations to be solved for y1, y2,...ym.

y is characterized by the function mapping $y : \mathbf{R} \rightarrow \mathbf{R}^m \,\!$.

r(x) is called a source term in x, and A(x) is an arbitrary function, both assumed continuous in x on defined intervals.

Characteristic Properties Differential equation
Implicit system of dimension m $y : \mathbf{R} \rightarrow \mathbf{R}^m \,\!$
$F : \mathbf{R}^{mn+2} \rightarrow \mathbf{R} \,\!$
$F \left ( x, y, \frac{\mathrm{d}y}{\mathrm{d}x}, \frac{\mathrm{d}^2y}{\mathrm{d}x^2}, \ldots, \frac{\mathrm{d}^ny}{\mathrm{d}x^n} \right ) = 0 \,\!$
Explicit system of dimension m $y : \mathbf{R} \rightarrow \mathbf{R}^m \,\!$

$F : \mathbf{R}^{mn+1} \rightarrow \mathbf{R} \,\!$

$\frac{\mathrm{d}^n y}{\mathrm{d}x^n} = F \left ( x, y, \frac{\mathrm{d}y}{\mathrm{d}x}, \frac{\mathrm{d}^2y}{\mathrm{d}x^2}, \ldots, \frac{\mathrm{d}^{n-1}y}{\mathrm{d}x^{n-1}} \right ) \,\!$
Autonomous: No x dependence $F \left ( y, \frac{\mathrm{d}y}{\mathrm{d}x}, \frac{\mathrm{d}^2y}{\mathrm{d}x^2}, \ldots, \frac{\mathrm{d}^ny}{\mathrm{d}x^n} \right ) = 0 \,\!$
Linear: nth derivative can be written

as a linear combination of the other

derivatives

$\frac{\mathrm{d}^n y}{\mathrm{d}x^n} = r(x) + \sum_{\alpha=0}^{n-1} \left ( A_\alpha (x) \frac{\mathrm{d}^\alpha y}{\mathrm{d}x^\alpha} \right ) \,\!$
Homogeneous: Source term

is zero

$r(x)=0 \,\!$

Notice the mapping from $\mathbf{R}^{mn+2} \,\!$ or $\mathbf{R}^{mn+1} \,\!$ corresponds to the map from x, y, and the (n+2) or (n+1) derivatives of y to the solution, in general implicit.

Examples

In the first group of examples, let u be an unknown function of x, and c and ω are known constants.

• Inhomogeneous first-order linear constant coefficient ordinary differential equation:
$\frac{du}{dx} = cu+x^2.$
• Homogeneous second-order linear ordinary differential equation:
$\frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0.$
• Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:
$\frac{d^2u}{dx^2} + \omega^2u = 0.$
• First-order nonlinear ordinary differential equation:
$\frac{du}{dx} = u^2 + 1.$
• Second-order nonlinear ordinary differential equation describing the motion of a pendulum of length L:
$L\frac{d^2u}{dx^2} + g\sin u = 0.$

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

• Homogeneous first-order linear partial differential equation:
$\frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0.$
• Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.$
$\frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}.$

Related concepts

• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.

Connection to difference equations

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.

Universality of mathematical description

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black–Scholes equation in finance is for instance, related to the heat equation.

Exact solutions

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

In the table below, H(x), Z(x), H(y), Z(y), or H(x,y), Z(x,y) are any integrable functions of x or y (or both), and A, B, C, I, L, N, M are all constants. In general A, B, C, I, L, are real numbers, but N, M, P and Q may be complex. The differential equations are in their equivalent and alternative forms which lead to the solution through integration.

Differential equation General solution
1 $\frac{\mathrm{d}y}{\mathrm{d}x} = F(x)\,\!$

$\mathrm{d}y= F(x) \, \mathrm{d}x\,\!$

$y=\int F(x) \, \mathrm{d}x\,\!$
2 $\frac{\mathrm{d}y}{\mathrm{d}x} = F(y)\,\!$

$\mathrm{d}y= F(y) \, \mathrm{d}x\,\!$

$x=\int \frac{\mathrm{d}y}{F(y)}\,\!$
3 $H(y)\frac{\mathrm{d}y}{\mathrm{d}x} + Z(x)= 0\,\!$

$H(y) \, \mathrm{d}y+ Z(x) \, \mathrm{d}x =0\,\!$

$\int H(y) \, {\mathrm{d}y}+\int Z(x) \, {\mathrm{d}x} = C\,\!$
4 $\frac{\mathrm{d}y}{\mathrm{d}x} + H(x)y+Z(x)= 0\,\!$

$\mathrm{d}y + H(x)y \, \mathrm{d}x+Z(x) \, \mathrm{d}x= 0\,\!$

$y = - e^{- U(x)} \int e^{U(x)}Z(x) \, {\mathrm{d}x}\,,$
where    $U(x) = \int H(x) \, \mathrm{d}x\,\!$
5 $\frac{\mathrm{d}y}{\mathrm{d}x} = F \left( \frac{y}{x} \right ) \,\!$ $\ln Cx = \int \frac{ \mathrm{d} r}{F(r) - r} \, \!$

solution may be implicit in x and y, obtained by calculating above integral then using

back-substituting $r = y/x \,\!$

6 $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = F(y) \,\!$ $x = \pm \int \frac{ \mathrm{d} y}{\sqrt{2 \int F(y) \, \mathrm{d} y + C_1}} + C_2 \, \!$
7 $H(x,y) \frac{\mathrm{d}y}{\mathrm{d}x} + Z(x,y) = 0 \,\!$

$H(x,y) \, {\mathrm{d}y} + Z(x,y) \, {\mathrm{d}x} = 0 \,\!$

If DE is exact so that $\frac{\partial H}{\partial x} = \frac{\partial Z}{\partial y} \, \!$

then the solution is:

$F(x,y) = \int \left [ H(x,y) \, \mathrm{d} y + Z(x,y) \, \mathrm{d} x \right ] + \gamma (y) + \chi (x) = C \, \!$

where $\gamma (y) \,\!$ and $\chi (x) \,\!$ are constant functions from the integrals rather than constant values, which are

set to make the final function $F(x,y) \,\!$ hold.

If DE is not exact, the functions H(x,y) or Z(x,y) may be used to obtain an integrating factor, the DE is

then multiplied through by that factor and the solution proceeds the same way.

8 $\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + I\frac{\mathrm{d}y}{\mathrm{d}x} + Ly = 0\,\!$

If $I^2 > 4L\,\!$

then $y=Ne^{ \left ( -I+\sqrt{I^2 - 4L} \right )\frac{x}{2}} + Me^{-\left ( I+\sqrt{I^2 - 4L} \right )\frac{x}{2}}\,\!$

If $I^2 = 4L\,\!$

then $y = (Ax + B)e^{-Ix/2}\,\!$

If $I^2 < 4L\,\!$

then $y = e^{ -I\frac{x}{2}} \left [ P \sin{\left ( \sqrt{\left | I^2-4L \right |}\frac{x}{2} \right )} + Q\cos{\left ( \sqrt{\left | I^2-4L \right |}\frac{x}{2} \right )} \right ] \,\!$

9 $\sum_{\alpha=1}^d I_{\alpha} \frac{\mathrm{d}^\alpha y}{\mathrm{d}x^\alpha} = 0\,\!$

$\sum_{\alpha=1}^d A_{\alpha} e^{B_{\alpha} x} = 0 \,\!$

where $B_{\alpha} \,\!$ are the d solutions of the polynomial of degree d:

$\prod_{\alpha=1}^d \left ( B - B_{\alpha} \right ) = 0 \,\!$

Note that 3 and 4 are special cases of 7, they are relatively common cases and included for completeness.

Similarly 8 is a special case of 9, but 8 is a relatively common form, particularly in simple physical and engineering problems.

Notable differential equations

Biology

• Verhulst equation – biological population growth
• von Bertalanffy model – biological individual growth
• Lotka–Volterra equations – biological population dynamics
• Replicator dynamics – may be found in theoretical biology
• Hodgkin-Huxley model - neural action potentials

References

• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
• A. D. Polyanin 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.
• W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection
• E.L. Ince, Ordinary Differential Equations, Dover Publications, 1956
• E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955
• P. Blanchard, R.L. Devaney, G.R. Hall, Differential Equations, Thompson, 2006
• Calculus, Teach Yourself, P.Abbott and H. Neill, 2003 pages 266-277
• Further Elementary Analysis, R.I.Porter, 1978, chapter XIX Differential Equations</ref>

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