Method for solving first order and Bernoulli's differential equations

History

I came across the method concerning this article in an old math book from Doctor Granville (Elements of differential and integral calculus - ISBN-13: 978-968-18-1178-5). It doesn't appear to be a popular technique as when using it for my assignments I always had to explain what I was doing. As of yet, I still haven't found another text referencing it, which is why I decided to include it in my website.

In the original book this procedure is shown but never really explained, it is left as a sort of "it just works" thing. Here is my attempt to it clear.

Theory

Throughout this article we'll consider first order differential equations with function coefficients just as a special case of the Bernoulli's differential equation with .

Consider now the following ODE:

(1)

let be the product of two arbitrary functions and such that

we now restrict to be the solution of the ODE

(2)

with this it is possible to solve for by integrating

(3)

using we solve for by replacing inside the original ODE

the general solution to our original ODE can be simply obtained by multiplying and .

Comments

This method, while functional, may not always be the most practical. In some cases the differential equations for and may not have closed algebraic solutions. A more traditional substitution may in some situations also be easier than this method. Like always it is up to one to know which tool to apply for a given problem.

Example

Let's solve an example to show the method in practice

solve now for

replace in the equation

finally with and get

(4)