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^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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