Overview
Integrating Factors is a method to solve a first order equation, given in standard form here
{% \frac{dy}{dx} + p(x)y = f(x) %}
Method of Solution
The integrating factor method tries to find a function {% u(x) %} such that
{% \frac{d}{dx}[u(x)y] = u(x) \frac{dy}{dx} + u(x)p(x)y %}
The
Product Rule
then implies that
{% \frac{d}{dx}[u(x)y] = u\frac{dy}{dx} + \frac{du}{dx}y %}
which means that
{% \frac{du}{dx} = u(x)p(x) %}
The method of
separation of variables
can be applied
{% \frac{du}{u} = p(x)dx %}
One solution is
{% u(x) = exp[\int p(x) dx] %}
Then multiplying the
standard form equation by {% u(x) %}
{% u(x) \frac{dy}{dx} + p(x) u(x) y = u(x) f(x) %}
{% \frac{d}{dx}[exp(\int p(x) dx)y] = exp(\int p(x) dx) f(x) %}
Integrating this equation gives
{% exp(\int p(x) dx)y = \int exp(\int p(x) dx) f(x) + C %}
Example
{% \frac{dy}{dx} + \frac{3y}{x} = \frac{e^{x}}{x^3} %}
The integrating factor is
{% exp[\int \frac{3}{x} dx] %}
{% \int \frac{3}{x} dx = 3 log x = log x^3 %}
{% exp[\int \frac{3}{x} dx] = x^3 %}
Multiplying both sides of the differential equation by the integrating factor we get
{% x^3 \frac{dy}{dx} + 3x^2y = e^x %}
This can then be integrated to get
{% x^3y = e^x + C %}
{% y = \frac{e^x + C}{x^3} %}