In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentialss. It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when 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.

Use

An integrating factor is any expression that a differential equation is multiplied by to facilitate integration. For example, the nonlinear second order equation

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,

is a constant.

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. This same method is used to solve the period of a simple pendulum.

Solving first order linear ordinary differential equations

Integrating factors are useful for solving ordinary differential equations that can be expressed in the form

, called the "integrating factor", which we can multiply through our differential equation in order to bring the left-hand side under a common derivative. For the canonical first-order linear differential equation.

refers to the sign function, and a logarithm in the antiderivative only appears when the original function involved a logarithm or a reciprocal (neither of which are defined for 0), such an interval will be the interval of validity of our solution.

transforms a non-integrable expression into an integrable derivative, then:

, which is a separable differential equation:

gives

By applying the product rule

We use this fact to simplify our expression to

is a constant.

Moving the exponential to the right-hand side, the general solution to ordinary differential equation is:

In the case of a homogeneous differential equation and the general solution to ordinary differential equation is:

1. .

for example, consider the differential equation

we obtain

The above equation can be rewritten as

By integrating both sides with respect to x we obtain

or

The same result may be achieved using the following approach

Reversing the quotient rule gives

or

or

is a constant.

Solving second order linear ordinary differential equations

to work as an integrating factor, then

for the integrating factor to be usable.

Example 1

For example, the differential equation

. This gives us

which can be rearranged to give

Integrating twice yields

Dividing by the integrating factor gives:

Example 2

A slightly less obvious application of second order integrating factors involves the following differential equation:

. However,

and from the Pythagorean identity relating cotangent and cosecant,

and can use integrating factors.

gives

which rearranged is

Integrating twice gives

Finally, dividing by the integrating factor gives

Solving nth order linear differential equations

th order differential equation and combine like terms. This will yield an equation in the form

times, dividing by the integrating factor on both sides to achieve the final result.

Example

A third order usage of integrating factors gives

thus requiring our equation to be in the form

For example in the differential equation

. Rearranging gives

Integrating thrice and dividing by the integrating factor yields

See also

• Variation of parameters

• Differential equations

• Product rule

• Quotient rule

• Exact differential

• Matrix exponential

References

• .

de:Exakte Differentialgleichung