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[parent] Bernoulli equation (Derivation)

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).



"Bernoulli equation" is owned by pahio.
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Other names:  Bernoulli differential equation

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Cross-references: general solution, linear differential equation of first order, Transforms, equation, real functions, continuous
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This is version 5 of Bernoulli equation, born on 2005-05-09, modified 2007-01-23.
Object id is 7032, canonical name is BernoulliEquation.
Accessed 13621 times total.

Classification:
AMS MSC: 34C05 (Ordinary differential equations :: Qualitative theory :: Location of integral curves, singular points, limit cycles)
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