Bernoulli Differential Equation

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Theorem

Bernoulli's differential eqn is a first order ordinary differential equation which can be put into the form:

$\displaystyle \frac {dy}{dx} + P \left({x}\right) y = Q \left({x}\right) y^n$

where $n \ne 0$ and $n \ne 1$.

It has the general solution:

$\displaystyle \frac {\mu \left({x}\right)} {y^{n-1}} = \int \left({1-n}\right) Q \left({x}\right) \mu \left({x}\right) dx + C$

where:

$\mu \left({x}\right) = e^{\int \left({1-n}\right) P \left({x}\right) dx}$

Make the substitution:

$z = y^{1-n}$

in the original eqn.

Then we have:

$\displaystyle $	$\displaystyle \frac {dz} {dy}$	$=$	$\displaystyle \left({1-n}\right) y^{-n}$	$\displaystyle $	Power Rule for Derivatives
$\displaystyle \implies$	$\displaystyle \frac {dz} {dy} \frac {dy} {dx} + P \left({x}\right) y \left({1-n}\right) y^{-n}$	$=$	$\displaystyle Q \left({x}\right) y^n \left({1-n}\right) y^{-n}$	$\displaystyle $
$\displaystyle \implies$	$\displaystyle \frac {dz} {dx} + \left({1-n}\right) P \left({x}\right) y^{1-n}$	$=$	$\displaystyle \left({1-n}\right) Q \left({x}\right)$	$\displaystyle $	Chain Rule
$\displaystyle \implies$	$\displaystyle \frac {dz} {dx} + \left({1-n}\right) P \left({x}\right) z$	$=$	$\displaystyle \left({1-n}\right) Q \left({x}\right)$	$\displaystyle $

$\mu \left({x}\right) = e^{\int \left({1-n}\right) P \left({x}\right) dx}$

and this can be used to obtain:

$\mu \left({x}\right) z = \int \left({1-n}\right) Q \left({x}\right) \mu \left({x}\right) dx + C$

Substituting $\displaystyle z = y^{1-n} = \frac 1 {y^{n-1}}$ finishes it off.

$\blacksquare$

When $n = 0$ or $n = 1$ the equation is already linear, and the technique for solving that can be used.

This entry was named for Jacob Bernoulli.

\(\displaystyle \)	\(\displaystyle \frac {dz} {dy}\)	\(=\)	\(\displaystyle \left({1-n}\right) y^{-n}\)	\(\displaystyle \)	Power Rule for Derivatives
\(\displaystyle \implies\)	\(\displaystyle \frac {dz} {dy} \frac {dy} {dx} + P \left({x}\right) y \left({1-n}\right) y^{-n}\)	\(=\)	\(\displaystyle Q \left({x}\right) y^n \left({1-n}\right) y^{-n}\)	\(\displaystyle \)
\(\displaystyle \implies\)	\(\displaystyle \frac {dz} {dx} + \left({1-n}\right) P \left({x}\right) y^{1-n}\)	\(=\)	\(\displaystyle \left({1-n}\right) Q \left({x}\right)\)	\(\displaystyle \)	Chain Rule
\(\displaystyle \implies\)	\(\displaystyle \frac {dz} {dx} + \left({1-n}\right) P \left({x}\right) z\)	\(=\)	\(\displaystyle \left({1-n}\right) Q \left({x}\right)\)	\(\displaystyle \)