Question

can someone explain how to solve bernoulli differential equations?

Answer 1

You use Bernoulli in situations like this
\[y'=p(x)y+q(x)y^n\]

Answer 2

What you do is divide everything by y^n
\[\frac{y'}{y^n}=p(x)\frac{y}{y^n}+q(x)\]
\[\frac{y'}{y^n}=p(x)y^{1-n}+q(x)\]

Answer 3

Then use substitution
\[v=y^{1-n}\]
\[(1-n) y^{-n}\]

Answer 4

Now plug in back in to equation
\[\frac{(v')}{1-n}=p(x)v+q(x)\]

Answer 5

Now it is linear differential equation from here

Answer 6

Btw,\[v'=(1-n) y^{-n}\]