PlanetMath: Bernoulli equation

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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 $$F = \lambda_1v+\lambda_2v^k.$$ 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 $$\frac{dz}{dx}+(-k+1)f(x)z = (-k+1)g(x)$$ 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, substitution, equation, real functions, continuous
There are 2 references to this entry.

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 18111 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

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