Bernoulli equation and its physical applications

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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