# Bernoulli equation

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 ($\neq 0$,  $\neq 1$).  Such a nonlinear equation (http://planetmath.org/DifferentialEquation) is got e.g. in examining the motion of a 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).

## References

• 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele.  – Kirjastus Valgus, Tallinn (1966).
Title Bernoulli equation BernoulliEquation 2013-03-22 15:15:03 2013-03-22 15:15:03 pahio (2872) pahio (2872) 11 pahio (2872) Derivation msc 34C05 Bernoulli differential equation RiccatiEquation