Talk:PlanetPhysics/Bernoulli Equation and its Physical Applications

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The {\em Bernoulli equation} has the form

\begin{align} \frac{dy}{dx}+f(x)y = g(x)y^k \end{align} where $f$ and $g$ are continuous real \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} and $k$ is a constant ($\neq 0$, \,$\neq 1$).\, Such an equation is got e.g. in examining the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} 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{align} y^{-k}\frac{dy}{dx}+f(x)y^{-k+1} = g(x). \end{align} The substitution \begin{align} z := y^{-k+1} \end{align} transforms (2) into $$\frac{dz}{dx}+(-k+1)f(x)z = (-k+1)g(x)$$ which is a linear \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} 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).

\begin{thebibliography}{9} \bibitem{NP}{\sc N. Piskunov:} {\em Diferentsiaal- ja integraalarvutus k\~{o}rgematele tehnilistele \~{o}ppeasutustele}. \,-- Kirjastus Valgus, Tallinn (1966). \end{thebibliography}

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