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| Meta Title | Partial differential equation | Solutions of PDEs, Partial Integrals & Variables | Britannica |
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| Boilerpipe Text | partial differential equation
, in
mathematics
,
equation
relating a
function
of several variables to its partial
derivatives
. A
partial derivative
of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held constant (
compare
ordinary differential equation
). The partial
derivative
of a function is again a function, and, if
f
(
x
,
y
) denotes the original function of the variables
x
and
y
, the partial derivative with respect to
x
—i.e., when only
x
is allowed to vary—is typically written as
f
x
(
x
,
y
) or ∂
f
/∂
x
. The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a second-order partial derivative. For example, taking the partial derivative of
f
x
(
x
,
y
) with respect to
y
produces a new function
f
x
y
(
x
,
y
), or ∂
2
f
/∂
y
∂
x
. The order and degree of partial
differential
equations are defined the same as for ordinary differential equations.
In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.
Many physically important partial differential equations are second-order and linear. For example:
More From Britannica
analysis: Partial differential equations
u
x
x
=
u
t
(one-dimensional
heat
equation)
u
x
x
−
u
y
y
= 0 (one-dimensional
wave
equation)
The behaviour of such an equation depends heavily on the coefficients
a
,
b
, and
c
of
a
u
x
x
+
b
u
x
y
+
c
u
y
y
. They are called elliptic, parabolic, or hyperbolic equations according as
b
2
− 4
a
c
< 0,
b
2
− 4
a
c
= 0, or
b
2
− 4
a
c
> 0, respectively. Thus, the Laplace equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.
This article was most recently revised and updated by
William L. Hosch
.
AI-generated answers
from Britannica articles. AI makes mistakes, so verify using Britannica articles. |
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**partial differential equation**, in [mathematics](https://www.britannica.com/science/mathematics), [equation](https://www.britannica.com/science/equation) relating a [function](https://www.britannica.com/science/function-mathematics) of several variables to its partial [derivatives](https://www.britannica.com/science/derivative-mathematics). A [partial derivative](https://www.britannica.com/science/partial-derivative) of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held constant (*compare* [ordinary differential equation](https://www.britannica.com/science/ordinary-differential-equation)). The partial [derivative](https://www.britannica.com/dictionary/derivative) of a function is again a function, and, if *f*(*x*, *y*) denotes the original function of the variables *x* and *y*, the partial derivative with respect to *x*—i.e., when only *x* is allowed to vary—is typically written as *f**x*(*x*, *y*) or ∂*f*/∂*x*. The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a second-order partial derivative. For example, taking the partial derivative of *f**x*(*x*, *y*) with respect to *y* produces a new function *f**x**y*(*x*, *y*), or ∂2*f*/∂*y*∂*x*. The order and degree of partial [differential](https://www.britannica.com/dictionary/differential) equations are defined the same as for ordinary differential equations.
In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.
Many physically important partial differential equations are second-order and linear. For example:
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[ More From Britannica analysis: Partial differential equations](https://www.britannica.com/science/analysis-mathematics/Partial-differential-equations#ref732535)
- *u**x**x* + *u**y**y* = 0 (two-dimensional [Laplace equation](https://www.britannica.com/science/Laplaces-equation))
- *u**x**x* = *u**t* (one-dimensional [heat](https://www.britannica.com/science/heat) equation)
- *u**x**x* − *u**y**y* = 0 (one-dimensional [wave](https://www.britannica.com/science/wave-physics) equation)
The behaviour of such an equation depends heavily on the coefficients *a*, *b*, and *c* of *a**u**x**x* + *b**u**x**y* + *c**u**y**y*. They are called elliptic, parabolic, or hyperbolic equations according as *b*2 − 4*a**c* \< 0, *b*2 − 4*a**c* = 0, or *b*2 − 4*a**c* \> 0, respectively. Thus, the Laplace equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.
This article was most recently revised and updated by [William L. Hosch](https://www.britannica.com/editor/William-L-Hosch/6481).
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- What is a differential equation?
- How is an ordinary differential equation different from other equations?
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**differential equation**, mathematical statement containing one or more [derivatives](https://www.britannica.com/science/derivative-mathematics)—that is, terms representing the rates of change of continuously varying quantities. [Differential](https://www.britannica.com/dictionary/Differential) equations are very common in science and [engineering](https://www.britannica.com/technology/engineering), as well as in many other fields of quantitative study, because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a [function](https://www.britannica.com/science/function-mathematics) that can be used to predict the behaviour of the original system, at least within certain constraints.
Differential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are [ordinary differential equations](https://www.britannica.com/science/ordinary-differential-equation) and [partial differential equations](https://www.britannica.com/science/partial-differential-equation). When the function involved in the [equation](https://www.britannica.com/science/equation) depends on only a single variable, its [derivatives](https://www.britannica.com/dictionary/derivatives) are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables, so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation. The following are examples of ordinary differential equations: 
In these, *y* stands for the function, and either *t* or *x* is the independent variable. The symbols *k* and *m* are used here to stand for specific constants.
[ More From Britannica analysis: Newton and differential equations](https://www.britannica.com/science/analysis-mathematics/Ordinary-differential-equations#ref732255)
Whichever the type may be, a differential equation is said to be of the *n*th order if it involves a [derivative](https://www.britannica.com/science/derivative-mathematics) of the *n*th order but no [derivative](https://www.britannica.com/dictionary/derivative) of an order higher than this. The equation  is an example of a partial differential equation of the second order. The theories of ordinary and partial differential equations are markedly different, and for this reason the two categories are treated separately.
Instead of a single differential equation, the object of study may be a simultaneous system of such equations. The formulation of the laws of [dynamics](https://www.britannica.com/science/dynamics-physics) frequently leads to such systems. In many cases, a single differential equation of the *n*th order is advantageously replaceable by a system of *n* [simultaneous equations](https://www.britannica.com/science/system-of-equations), each of which is of the first order, so that techniques from [linear algebra](https://www.britannica.com/science/linear-algebra) can be applied.
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[order](https://www.britannica.com/science/order-of-a-differential-equation)
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[See all related content](https://www.britannica.com/facts/differential-equation)
An ordinary differential equation in which, for example, the function and the independent variable are denoted by *y* and *x* is in effect an [implicit](https://www.merriam-webster.com/dictionary/implicit) summary of the essential characteristics of *y* as a function of *x*. These characteristics would presumably be more accessible to analysis if an explicit formula for *y* could be produced. Such a formula, or at least an equation in *x* and *y* (involving no derivatives) that is deducible from the differential equation, is called a solution of the differential equation. The process of deducing a solution from the equation by the applications of [algebra](https://www.britannica.com/science/algebra) and [calculus](https://www.britannica.com/science/calculus-mathematics) is called solving or [integrating](https://www.britannica.com/science/integration-mathematics) the equation. It should be noted, however, that the differential equations that can be explicitly solved form but a small minority. Thus, most functions must be studied by indirect methods. Even its existence must be proved when there is no possibility of producing it for inspection. In practice, methods from [numerical analysis](https://www.britannica.com/science/numerical-analysis), involving computers, are employed to obtain useful approximate solutions.
This article was most recently revised and updated by [William L. Hosch](https://www.britannica.com/editor/William-L-Hosch/6481).
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| Readable Markdown | **partial differential equation**, in [mathematics](https://www.britannica.com/science/mathematics), [equation](https://www.britannica.com/science/equation) relating a [function](https://www.britannica.com/science/function-mathematics) of several variables to its partial [derivatives](https://www.britannica.com/science/derivative-mathematics). A [partial derivative](https://www.britannica.com/science/partial-derivative) of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held constant (*compare* [ordinary differential equation](https://www.britannica.com/science/ordinary-differential-equation)). The partial [derivative](https://www.britannica.com/dictionary/derivative) of a function is again a function, and, if *f*(*x*, *y*) denotes the original function of the variables *x* and *y*, the partial derivative with respect to *x*—i.e., when only *x* is allowed to vary—is typically written as *f**x*(*x*, *y*) or ∂*f*/∂*x*. The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a second-order partial derivative. For example, taking the partial derivative of *f**x*(*x*, *y*) with respect to *y* produces a new function *f**x**y*(*x*, *y*), or ∂2*f*/∂*y*∂*x*. The order and degree of partial [differential](https://www.britannica.com/dictionary/differential) equations are defined the same as for ordinary differential equations.
In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.
Many physically important partial differential equations are second-order and linear. For example:
[ More From Britannica analysis: Partial differential equations](https://www.britannica.com/science/analysis-mathematics/Partial-differential-equations#ref732535)
- *u**x**x* = *u**t* (one-dimensional [heat](https://www.britannica.com/science/heat) equation)
- *u**x**x* − *u**y**y* = 0 (one-dimensional [wave](https://www.britannica.com/science/wave-physics) equation)
The behaviour of such an equation depends heavily on the coefficients *a*, *b*, and *c* of *a**u**x**x* + *b**u**x**y* + *c**u**y**y*. They are called elliptic, parabolic, or hyperbolic equations according as *b*2 − 4*a**c* \< 0, *b*2 − 4*a**c* = 0, or *b*2 − 4*a**c* \> 0, respectively. Thus, the Laplace equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.
This article was most recently revised and updated by [William L. Hosch](https://www.britannica.com/editor/William-L-Hosch/6481).
[AI-generated answers](https://www.britannica.com/about-britannica-ai) from Britannica articles. AI makes mistakes, so verify using Britannica articles. |
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