Can't Make Sense of Bernoulli Equation

[snowJT]

it...just...does...not...make...the...slightest...sense...to...me...

Here it goes...

$$y' + \frac{y}{x} = 3x^2y^2$$

This is a Bernoulli equation with $$P = \frac{1}{2}$$, $$Q = 3x^2$$, and $$n = 2$$. We first divide through by $$y^2$$, obtaining...

$$\frac{1}{y^2} \frac{dy}{dx} + \frac{y^-^1}{x} = 3x^2$$

We substitute $$z = y^-^1$$ and

$$\frac{dz}{dx} = -y^-^2\frac{dy}{dx} \Longrightarrow \frac{dy}{dx} = -y^2\frac{dz}{dx}$$

Substituting... $$-\frac{dz}{dx} + \frac{z}{x} = 3x^2 \Longrightarrow \frac{dz}{x} - \frac{z}{x} = -3x^2$$

I'll stop there... two things I don't get... how does $$P = \frac{1}{2}$$ and how does that substitution work...

If you can help, I would really appreciate it.

 

[HallsofIvy]

Yes, for that sentence to make any sense, you would have to know what a "Bernoulli equation" was, wouldn't you? Not remembering exactly and not having a differential equations textbook immediately handy, I "googled" on "Bernoulli differential equation" and get (from Wikipedia)
"an equation of the form y'+ P(x)y= Q(x)yⁿ for n not equal to 1"

There's obviously an error- either in your source or in your copying. That is a Bernoulli equation with P(x)= 1/x, not 1/2.

Dividing a Bernoulli equation by yⁿ and then making the substitution z= y^1-n always reduces the equation to a linear equation.

As for "how the substitution works" I don't know what more to say. It is shown pretty clearly there. What are you not understanding? (Other than that "P(x)= 1/2" should be P(x)= 1/x".)

Do you see how they go from z= y^-1 to dz/dx= -y^-2dy/dx? That's straight forward "chain rule". Then, of course, y^-2dy/dx in the differential equation becomes -dz/dx. y^-1/x is simply z/x. The differential equation y^-2dy/dx+ y^-1/x= 3x² becomes -dz/dx+ (1/x)z= 3x².