Bernoulli Differential Equation

A Bernoulli differential equation is a differential equation in the form

$$\frac{dy}{dx}+P(x)y=Q(x)y^{\alpha }$$

Where $n\in \mathbb{R}$

Note that this equation is linear when $\alpha \in \{0,1\}$

General Solution

1. divide by $y^{\alpha }$

$$y^{-\alpha }\frac{\partial y}{\partial x}+P(x)y^{1-\alpha }=Q(x)$$

2. Use the substitution

$$\text{let}v=y^{1-\alpha }\Rightarrow \frac{dv}{dx}=(1-\alpha )y^{-n}\frac{dy}{dx}$$ $$\Rightarrow \frac{dv}{dx}+P(x)v=Q(x)$$

3. use intergrating factor where $I=e^{\int P(x)dx}$

$$\begin{array}{r}\frac{d}{dx}(Iv)=Q(x)\\ Iv=\int Q(x)dx\end{array}$$

4. solve this equation