Integrating factor
Encyclopedia
In mathematics
, an integrating factor is a function
that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equation
s, but is also used within multivariable calculus
, in this case often multiplying through by an integrating factor allows an inexact differential
to be made into an exact differential
(which can then be integrated to give a scalar field
). This is especially useful in thermodynamics
where temperature becomes the integrating factor that makes entropy
an exact differential.
Consider a function
. We multiply both sides of (1) by 
We want the left hand side to be in the form of the derivative of a product (see product rule
), such that (2) can be written as
The left hand side in (3) can now be integrated
We can now solve for
Applying the product rule
to the left hand side of (3) and equating to the left hand side of (2)
From which it is clear that
obeys the differential equation :
Solving (5) gives
is called an integrating factor.
We can see that in this case
(Note we do not need to include the integrating constant - we need only a solution, not the general solution)
Multiplying both sides by
we obtain
or
which gives
admits
as an integrating factor:
To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule
:
Therefore
This form may be more useful, depending on application. Performing a separation of variables
will give:
this is an implicit
solution which involves a nonelementary integral
. Though likely too obscure to be useful, this is a general solution. Also, because the previous equation is first order, it could be used for numeric solution in favor of the original equation.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an integrating factor is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s, but is also used within multivariable calculus
Multivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
, in this case often multiplying through by an integrating factor allows an inexact differential
Inexact differential
An inexact differential or imperfect differential is a specific type of differential used in thermodynamics to express the path dependence of a particular differential. It is contrasted with the concept of the exact differential in calculus, which can be expressed as the gradient of another...
to be made into an exact differential
Exact differential
A mathematical differential is said to be exact, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q....
(which can then be integrated to give a scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
). This is especially useful in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
where temperature becomes the integrating factor that makes entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
an exact differential.
Use in solving first order linear ordinary differential equations
Consider an ordinary differential equation of the formConsider a function
. We multiply both sides of (1) by We want the left hand side to be in the form of the derivative of a product (see product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
), such that (2) can be written as
The left hand side in (3) can now be integrated
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
We can now solve for
Applying the product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
to the left hand side of (3) and equating to the left hand side of (2)
From which it is clear that
obeys the differential equation :Solving (5) gives
is called an integrating factor.Example
Solve the differential equationWe can see that in this case
(Note we do not need to include the integrating constant - we need only a solution, not the general solution)Multiplying both sides by
we obtainor
which gives
General use
The term "integrating factor" is synonymous with the solution of first order linear equations. One should bear in mind, however, that an integrating factor is any expression that a differential equation is multiplied by to facilitate integration, and that it is by no means restricted to first order linear equations. For example, the nonlinear second order equationadmits
as an integrating factor:To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
:
Therefore
This form may be more useful, depending on application. Performing a separation of variables
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
will give:
this is an implicit
Implicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
solution which involves a nonelementary integral
Nonelementary integral
In mathematics, a nonelementary antiderivative is an antiderivative for which it can be shown that there exists no formula in terms of elementary functions . A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist...
. Though likely too obscure to be useful, this is a general solution. Also, because the previous equation is first order, it could be used for numeric solution in favor of the original equation.
See also
- Method of variation of parametersMethod of variation of parametersIn mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations...
- Examples of differential equationsExamples of differential equationsDifferential equations arise in many problems in physics, engineering, and other sciences. The following examples show how to solve differential equations in a few simple cases when an exact solution exists....
- Product ruleProduct ruleIn calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
- Exact differentialExact differentialA mathematical differential is said to be exact, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q....