Integrating factor

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In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given ordinary differential equation.

Consider an ordinary differential equation of the form

$y'+a(x)y = b(x)\quad\quad\quad (1)$

where $y = y (x)$ is an unknown function of $x$ , and $a (x)$ and $b (x)$ are given functions.

The integrating factor method works by turning the left hand side into the form of the derivative of a product.

Consider a function $M (x)$ . We multiply both sides of (1) by $M (x):$

$M(x)y' + M(x)a(x)y = M(x)b(x).\quad\quad\quad (2)$

We want the left hand side to be in the form of the derivative of a product (see product rule). In fact, if we assume this the left hand side can be rearranged as

$(M(x)y)' = M(x)b(x).\quad\quad\quad (3)$

The left hand side in (3) can now be integrated much more easily by means of the fundamental theorem of calculus,

$y(x) M(x) = \int b(x) M(x)\,dx + C,$

where $C$ is a constant (see arbitrary constant of integration). We can now solve for $y (x),$

$y(x) = \frac{\int b(x) M(x)\, dx + C}{M(x)}.\,$

However, to explicitly solve for $y (x)$ we need to find an expression for $M (x).$

Rewrite (3) using the product rule.

$(M(x)y)' = M'(x)y + M(x)y' = M(x)b(x).\quad\quad\quad$

Identify terms in (2) and it's clear that $M (x)$ obeys the differential equation :

$M'(x) = a(x)M(x) .\quad\quad\quad (4)\,$

To get $M (x)$ , divide both sides by $M (x):$

$\frac{M'(x)}{M(x)}-a(x) = 0.\quad\quad\quad (5)$

Equation (5) is now in the form of a logarithmic derivative. Solving (5) gives

$M(x)=e^{\int a(x)\,dx}.$

We see that multiplying by $M (x)$ and the property $M'(x) = a (x) M (x)$ were essential in solving this differential equation. $M (x)$ is called an integrating factor. The name comes from the fact that it is an integral, and it shows as a multiple in the equation (hence a factor).