PlanetPhysics/Bernoulli Equation and its Physical Applications

The Bernoulli equation has the form

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

where $f$ and $g$ are continuous real [[../Bijective/|functions]] and $k$ is a constant ($\ne 0$, \,$\ne 1$).\, Such an equation is got e.g. in examining the [[../CosmologicalConstant/|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

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

The substitution

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

transforms (2) into $\frac{dz}{dx}+(-k+1)f(x)z=(-k+1)g(x)$ which is a linear [[../DifferentialEquations/|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).