Bernoulli's Equation

y^{'} + P (x) y = Q (x) y^{n}

Solving

Change variables: $v (x) = y^{1 - n}, \frac{d v}{d x} = (1 - n) y^{- n} \frac{d y}{d x}$
Differentiate both sides wrt $x$ where $y = y (x)$ , and simplify $$\frac{1}{1-n}\frac{dv}{dx} = \frac{1}{y^n}\frac{dy}{dx}$$
Divide both sides of the original ODE by $y^{n}$ , simplify $$\frac{1}{y^n}\frac{dy}{dx} + P(x) y^{1-n} = Q(x)$$
Substitute with change to $v$ variable to obtain first order ODE
Solve ODE for $v (x)$
Substitute $v = y^{1 - n}$ to return to original variables $y (x)$