Bernoulli Differential Equation

$${dy \over dx} + p(x)y = q(x)y^n.$$ (1)

Let $v \equiv y^{1-n}$ for $n\not = 1$, then

$${dv\over dx} = (1-n)y^{-n}{dy\over dx}.$$ (2)

Rewriting (1) gives

$$y^{-n}{dy \over dx}= q(x)-p(x)y^{1-n} = q(x)-vp(x).$$ (3)

Plugging (3) into (2),

$${dv\over dx} = (1-n)[q(x)-vp(x)].$$ (4)

Now, this is a linear First-Order Ordinary Differential Equation of the form

$${dv\over dx}+vP(x)=Q(x),$$ (5)

where $P(x)\equiv (1-n)p(x)$ and $Q(x)\equiv (1-n)q(x)$. It can therefore be solved analytically using an Integrating Factor

$$v = {{\int e^{\int P(x)\,dx} Q(x)\,dx + C}\over e^{\int P(x)\,dx}}$$

$$= {{(1-n)\int e^{(1-n)\int p(x)\,dx} q(x)\,dx + C}\over e^{(1-n)\int p(x)\,dx}},$$ (6)

where $C$ is a constant of integration. If $n=1$, then equation (1) becomes

$${dy \over dx}=y(q-p)$$ (7)

$${dy\over y}=(q-p)\,dx$$ (8)

$$y=C_2 e^{\int [q(x)-p(x)]\,dx}.$$ (9)

The general solution is then, with $C_1$ and $C_2$ constants,

$$y = \cases{ \left[{{{(1-n)\int e^{(1-n)\int p(x)\,dx} q(x)\... ...or $n\not=1$\cr C_2 e^{\int [q(x)-p(x)]\,dx} & for $n=1$.\cr}$$ (10)