Integrating Factor Method

We got introduced to the concept of differential equations . In this chapter we will learn about a specific method to solve differential equations called the Integrating Factor Method.

I. Introduction

This method is used to solve linear, first order, differential equations. The equations that can be solved using this method are of the form:

$\dfrac{dy}{dx} + f(x)y = g(x)$

where $f(x)$ and $g(x)$ are integrable functions in the interval of integration.

II. Method

For any differential equation of the above form, if we take the integrating factor as:

$e^{\int f(x)dx}$ then the solution is:

$\displaystyle ye^{\int f(x)dx} = \int g(x) \cdot e^{\int f(x)dx} dx + C$

III. Proof

$\dfrac{dy}{dx} + f(x)y = g(x)$

Multiplying both sides with the integrating factor $e^{\int f(x)dx}$ we get:

$e^{\int f(x)dx} \left( \dfrac{dy}{dx} + f(x)y \right) = e^{\int f(x)dx} g(x)$ $...eqn (i)$

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First we will show that the LHS of the above equation

is equal to $\dfrac{d}{dx}\left(y e^{\int f(x)dx} \right)$:

$\dfrac{d}{dx}\left(y e^{\int f(x)dx} \right)$

$= \left(e^{\int f(x)dx} \right) \dfrac{dy}{dx} + y \dfrac{d}{dx}\left(e^{\int f(x)dx} \right)$

$= e^{\int f(x)dx} \dfrac{dy}{dx} + y \cdot \dfrac{d \left(e^{\int f(x)dx} \right) }{d(\int f(x)dx)}\cdot \dfrac{d\left(\int f(x)dx\right)}{dx}$

$= e^{\int f(x)dx} \dfrac{dy}{dx} + y \left(e^{\int f(x)dx} \right) f(x)$

$= e^{\int f(x)dx} \left(\dfrac{dy}{dx} + y \cdot f(x) \right)$

Substituting this in $eqn\ (i)$ we get:

$\dfrac{d}{dx}\left(y e^{\int f(x)dx} \right) = g(x) e^{\int f(x)dx}$

Integrating both sides, we get:

$\displaystyle \int \dfrac{d}{dx}\left(y e^{\int f(x)dx} \right) dx = \int g(x) e^{\int f(x)dx} dx$

$\Rightarrow \displaystyle y e^{\int f(x)dx} = \int g(x) e^{\int f(x)dx} dx$