Bernoulli Differential Equation

[bdh2991]

Homework Statement

solve the given differential equation: xdy/dx + y = y^-2

Homework Equations

The Attempt at a Solution

I don't understand how to do these substitutions...i got n=-2 then u=y^3, du/dx=3y^2

from there i don't know where to place them

 

[vela]

It probably doesn't make sense partially because you didn't calculate du/dx correctly. You're differentiating with respect to x, not y, so you need to use the chain rule:
$$\frac{du}{dx} = 3y^2\frac{dy}{dx}$$

Now, first, divide the differential equation by x so that the coefficient of y' is 1. Then get rid of the y's from the righthand side by multiplying by y². You end up with
$$y^2\frac{dy}{dx} + \frac{1}{x} y^3 = \frac{1}{x} $$ Now write that in terms of u=y³ and u'.

 

[HallsofIvy]

You can also note that this is a separable equation:
$$x\frac{dy}{dx}+ y= y^{-2}$$
$$x\frac{dy}{dx}= -y+ y^{-2}$$
$$\frac{dy}{-y+ y^{-2}}= \frac{dx}{x}$$
$$\frac{y^2 dy}{1- y^3}= \frac{dx}{x}$$

To integrate on the left, let $u= 1- y^3$.