PlanetPhysics/Bernoulli Equation and its Physical Applications

The Bernoulli equation has the form

${\begin{matrix}{\frac {dy}{dx}}+f(x)y=g(x)y^{k}\end{matrix}}$

where $f$ and $g$ are continuous real functions and $k$ is a constant ($\neq 0$, \,$\neq 1$).\, Such an equation is got e.g. in examining the motion of a body when the resistance of medium depends on the velocity $v$ as $F=\lambda _{1}v+\lambda _{2}v^{k}.$ The real function $y$ can be solved from (1) explicitly.\, To do this, divide first both sides by $y^{k}$.\, It yields

${\begin{matrix}y^{-k}{\frac {dy}{dx}}+f(x)y^{-k+1}=g(x).\end{matrix}}$

The substitution

${\begin{matrix}z:=y^{-k+1}\end{matrix}}$

transforms (2) into ${\frac {dz}{dx}}+(-k+1)f(x)z=(-k+1)g(x)$ which is a linear differential equation of first order.\, When one has obtained its general solution and made in this the substitution (3), then one has solved the Bernoulli equation (1).