Bernoulli differential equation

Bernoulli differential equations are ordinary differential equations in the form

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

If $n=0$ or $n=1$ then it is linear. Otherwise it is non-linear, although they can be transformed into a first order linear differential equation by means of a substitution

$z=y^{1-n}$

This will then reduce the equation (after some manipulation and simplification) to a first order linear ODE which can be solved with an integrating factor.

For example:

${\frac {dy}{dx}}-2x^{-1}y=-x^{2}y^{2}$

$y^{-2}{\frac {dy}{dx}}-2x^{-1}y^{-1}=-x^{2}$

Now we can make the substitution

$w=y^{-1},{\frac {dw}{dx}}=-y^{-2}{\frac {dy}{dx}}$

This will give us:

${\frac {dw}{dx}}+2x^{-1}w=x^{2}$

This can now be solved the same way as a linear equation. The final answer is

$y={\frac {5x^{2}}{x^{5}+C}}$