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 $$F = \lambda_1v+\lambda_2v^k.$$ 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).