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Bernoulli equation and its physical applications (Topic)

The Bernoulli equation has the form

$\displaystyle \frac{dy}{dx}+f(x)y = g(x)y^k$ (1)
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

$\displaystyle F = \lambda_1v+\lambda_2v^k.$
The real function $y$ can be solved from (1) explicitly.  To do this, divide first both sides by $y^k$.  It yields
$\displaystyle y^{-k}\frac{dy}{dx}+f(x)y^{-k+1} = g(x).$ (2)
The substitution
$\displaystyle z := y^{-k+1}$ (3)
transforms (2) into

$\displaystyle \frac{dz}{dx}+(-k+1)f(x)z = (-k+1)g(x)$
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).

Bibliography

1
N. PISKUNOV: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele.  – Kirjastus Valgus, Tallinn (1966).



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Cross-references: differential equation, motion, functions

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Physics Classification: 02.30.Hq (Ordinary differential equations)
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