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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:

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

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

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