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Integrating Factor

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If a differential equation of the form

$\displaystyle p(x,y)dx + q(x,y)dy = 0 $

becomes exact by multiplication with a function $ \mu(x,y)$ then $ \mu(x,y)$ is called an integrating factor of the differential equation.

Note that it follows that

$\displaystyle (\mu p)_y = (\mu q)_x \ \,, $

is a necessary condition provided $ \mu $ is an integrating factor. This means that $ \mu $ satisfies the partial differential equation

$\displaystyle \mu_y p + \mu p_y = \mu_x q + \mu q_x \ .$

In general integrating factors are not easy to find. Sometimes it is possible to find an integrating factor $ \mu $ which depends only on $ x$ (or only on $ y$).

The condition on an integrating factor depending only on $ x$ is

$\displaystyle \frac{d}{dx} \operatorname{ln} \mu = \frac{\mu_x}{\mu} = \frac{p_y-q_x}{q} \ . $

This can be solved, if the right hand side $ \dps \frac{p_y - q_x}{q}$ is only a function of $ x .$ In this case it follows that

$\displaystyle \mu (x) = e^{\int \frac{p_y-q_x}{q} \ dx} \ . $

Similarly integrating factors depending only on $ y$ may be determined if they exist.

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Example:

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  automatically generated 7/ 5/2005