Thomas' Labs

Method for solving first order and Bernoulli's differential equations


This site as of now just a technology demonstration and its claims should not be taken as true (even though I myself am pretty confident they are)


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.


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 diff_8ad4c4398384fa93443b29d2a957ead413e739b7.svg.

Consider now the following ODE:


let diff_a32403c28532fbbd0022d96f4ac9217ed28d4783.svg be the product of two arbitrary functions diff_cae0b883f5cc644c1d045cfa3ec6c73d220b9431.svg and diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg such that


we now restrict diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg to be the solution of the ODE


with this it is possible to solve for diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg by integrating


using diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg we solve for diff_cae0b883f5cc644c1d045cfa3ec6c73d220b9431.svg by replacing diff_a32403c28532fbbd0022d96f4ac9217ed28d4783.svg inside the original ODE


the general solution to our original ODE can be simply obtained by multiplying diff_cae0b883f5cc644c1d045cfa3ec6c73d220b9431.svg and diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg.


This method, while functional, may not always be the most practical. In some cases the differential equations for diff_cae0b883f5cc644c1d045cfa3ec6c73d220b9431.svg and diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg 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.


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


solve now for diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg


replace diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg in the equation


finally with diff_cae0b883f5cc644c1d045cfa3ec6c73d220b9431.svg and diff_e47f5f95e112b5509fed2e5ff8ab337116e0c68f.svg get diff_a32403c28532fbbd0022d96f4ac9217ed28d4783.svg