Bernoulli's equation

 

정의1

$\frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y = f(x)y^{n}$     $(n \in \mathbb{R})$

is called Bernoulli's equation.

 

# If $n = 0$ or $n=1$, then $\frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y = f(x)y^{n}$ is a linear equation.

 

 

 

Solution

Idea : $u = y^{1-n}$  $\Rightarrow$  $y = u^{\frac{1}{1-n}}$  $\Rightarrow$  $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{1-n} u^{\frac{n}{1-n}} \frac{\mathrm{d}u}{\mathrm{d}x}$

 

$\frac{\mathrm{d}y}{\mathrm{d}x} + P(x)y = f(x)y^{n}$

     $\Rightarrow$  $\frac{1}{1-n} u^{\frac{n}{1-n}} \frac{\mathrm{d}u}{\mathrm{d}x} + P(x)u^{\frac{1}{1-n}} = f(x) u^{\frac{n}{1-n}}$

     $\Rightarrow$  $\frac{\mathrm{d}u}{\mathrm{d}x} + (1-n)P(x)u = (1-n)f(x)$

It is a linear equation.