Differential equations can be classified several different ways. Besides describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is ordinary or partial, linear or non-linear, and homogeneous or heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.
Ordinary differential equations An
ordinary differential equation (
ODE) is an equation containing an unknown
function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a
dependent variable (often denoted ), which, therefore,
depends on . Thus is often called the
independent variable of the equation. The term "
ordinary" is used in contrast with the term
partial differential equation, which may be with respect to
more than one independent variable. As, in general, the solutions of a differential equation cannot be expressed by a
closed-form expression,
numerical methods are commonly used for solving differential equations on a computer.
Partial differential equations A
partial differential equation (
PDE) is a differential equation that contains unknown
multivariable functions and their
partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or using a relevant
computer model. PDEs can be used to describe a wide variety of phenomena in nature such as
sound,
heat,
electrostatics,
electrodynamics,
fluid flow,
elasticity, or
quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional
dynamical systems, partial differential equations often model
multidimensional systems.
Stochastic partial differential equations generalize partial differential equations for modeling
randomness.
Linear differential equations Linear differential equations are differential equations that are
linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of
integrals. Many differential equations that are encountered in
physics are linear, for example ODEs describing
radioactive decay and PDEs for
heat transfer by thermal diffusion. These lead to
special functions, which may be defined as solutions of linear differential equations (see
Holonomic function).
Non-linear differential equations A
non-linear differential equation is a differential equation that is not a
linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular
symmetries. Nonlinear differential equations can exhibit very complicated behaviour over extended time intervals, characteristic of
chaos. Even the fundamental questions of existence and uniqueness of solutions for nonlinear differential equations 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). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution. In some circumstances, nonlinear differential equations may be approximated by linear ones. 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. Similarly, when a fixed point or stationary solution of a nonlinear differential equation has been found, investigation of its stability leads to a linear differential equation.
Equation order and degree The
order of the differential equation is the highest
order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only
first-order derivatives is a
first-order differential equation, an equation containing the
second-order derivative is a
second-order differential equation, and so on. When it is written as a
polynomial equation in the unknown function and its derivatives, the
degree of the differential equation is, depending on the context, the
polynomial degree in the highest derivative of the unknown function, or its
total degree in the unknown function and its derivatives. In particular, a
linear differential equation has degree one for both meanings, but the non-linear differential equation y'+y^2=0 is of degree one for the first meaning but not for the second one. Differential equations that describe natural phenomena usually have only first and second order derivatives in them, but there are some exceptions, such as the
thin-film equation, which is a fourth order partial differential equation.
Homogeneous linear equations A linear differential equation is
homogeneous if each term in the equation includes either the dependent variable or one of its derivatives. If this is not the case, so that there is a term that does not include either the dependent variable itself or a derivative of it, the equation is
inhomogeneous or
heterogeneous. See the examples section below.
Examples The first group of examples are ordinary differential equations, where
u is an unknown function of
x, and
c and
ω are constants that are assumed to be known. These examples illustrate the distinction between
linear and
nonlinear differential equations, and between homogeneous differential equations and
inhomogeneous ones, defined above. • 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 + 4. • Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a
pendulum of length
L: • : L\frac{d^2u}{dx^2} + g\sin u = 0. The next group of examples are partial differential equations. 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. • Third-order non-linear partial differential equation, the
KdV equation: • : \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. ==Initial conditions and boundary conditions==