Bernoulli and homogeneous equations.

Examples of Bernoulli and homogeneous equations.

These notes are set so that you get to prove the main results by solving smaller problems that when put together give the big result. The answers to the problems are in the videos. You will get the most out of these notes if you do (or try) the problems before looking at the videos.
A Bernoulli equation is one of the form $$ y' + p(x) y = f(x)y^n. $$ The method for solving these is to let $y = v^\alpha$ plug this into the equation and then choose $\alpha$ so that the equation becomes first order linear in the new variable $v$.

Problem: For the Bernoulli equation $$ y' -3 y = 24 y^3 $$ find both the general solution and the solution with $y(0)=1$.

Solution.

A homogeneous equation is one that can be written in the form $$ y' = F\left( \frac{y}{x} \right). $$ The trick on these to let $ v = \dfrac{y}{x} $, that is $ y = xv$. Then the equation becomes a separable equation in the variables $x$ and $v$.

Problem: Find the general solution to the equation $$ y' = \frac{2y^2+xy+x^2}{2xy+x^2} . $$

Solution.