7 Finding the integrating factor for linear ODEs

The differential equation governing the current i in a circuit with inductance L and resistance R in series subject to a constant applied electromotive force E cos ω t , where E and ω are constants, is

L d i d t + R i = E cos ω t (1)

This is an example of a linear differential equation in which i is the dependent variable and t is the independent variable. The general standard form of a linear first order differential equation is normally written with ‘ y ’ as the dependent variable and with ‘ x ’ as the independent variable and arranged so that the coefficient of d y d x is 1. That is, it takes the form:

d y d x + f ( x ) y = g ( x ) (2)

in which f ( x ) and g ( x ) are functions of x .

Comparing (1) and (2), x is replaced by t and y by i to produce d i d t + f ( t ) i = g ( t ) . The function f ( t ) is the coefficient of the dependent variable in the differential equation. We shall describe the method of finding the integrating factor for (1) and then generalise it to a linear differential equation written in standard form.

Step 1 Write the differential equation in standard form i.e. with the coefficient of the derivative equal to 1. Here we need to divide through by L :

d i d t + R L i = E L cos ω t .

Step 2 Integrate the coefficient of the dependent variable (that is, f ( t ) = R L ) with respect to the independent variable (that is, t ), and ignoring the constant of integration

R L d t = R L t .

Step 3 Take the exponential of the function obtained in Step 2.

This is the integrating factor (I.F.)

I.F. = e R t L .

This leads to the following Key Point on integrating factors:

Key Point 4

The linear differential equation (written in standard form):

d y d x + f ( x ) y = g ( x ) has an integrating factor I.F. = exp f ( x ) d x

Find the integrating factors for the equations

  1. x d y d x + 2 x y = x e 2 x
  2. t d i d t + 2 t i = t e 2 t
  3. d y d x ( tan x ) y = 1.
  1. Step 1 Divide by x to obtain d y d x + 2 y = e 2 x

    Step 2 The coefficient of the independent variable is 2 hence 2 d x = 2 x

    Step 3 I.F. = e 2 x

  2. The only difference from (1) is that i replaces y and t replaces x . Hence I.F. = e 2 t .
  3. Step 1 This is already in the standard form.

    Step 2 tan x d x = sin x cos x d x = ln cos x .

    Step 3 I.F. = e ln cos x = cos x