First Order Linear Differential Equations, Integrating Factor

In a Nut Shell: There are two common methods to solve first order, linear, ordinary

differential equations. They include:

Separation of variables (easiest if the dependent and independent variables separate.)

Find an integrating factor.

Review:

A differential equation, y ’ + p(x)y = q(x) is said to be linear if its

dependent variable y and its derivative y ’ are linear.

(Linear = No nonlinear terms involving y or y ‘)

Example of a first order linear, ordinary d.e. - separable form

dy/dx = y’ = f(x) This is the differential equation.

y(a) = A = constant = This is the initial condition on y the dependent variable

Solve by direct integration of dy = f(x) dx (separated form)

y(x) = ∫ f(x) dx + C where C is determined using the initial condition

Click here for another example using separation of variables.

Next look at the method of solution using an Integrating factor.

NOTE: Every first order, linear, ordinary d.e. has an integrating factor.

y ’ + p(x) y = q(x)

Integrating factor (IF) is as follows: IF = e ^{∫ p\(x) dx} Constant of integration

does not matter here.

e ^{∫ p(x) dx} y ’ + p(x) e ^{∫ p(x) dx} y = q(x) e ^{∫ p(x) dx}

d/dx [e ^{∫ p(x) dx} y ] = q(x) e ^{∫ p(x) dx}

e ^{∫ p(x) dx} y = ∫ [q(x) e ^{∫ p(x) dx} ] dx

Here the constant of

y(x) = e^{- ∫ p(x) dx} { q(x) e ^{∫ p(x) dx} }dx + C (solution) integration matters.

Click here to continue with an example.

Return to Differential Equations Notes

Copyright © 2017 Richard C. Coddington

All rights reserved.