Substitution for Differential equations Dear StackExchange users,
I have a little question ... I just don't have a clue how that works
I have the following differential equation
$$\frac{dy}{dx}=xy^2-2\frac{y}{x}-\frac{1}{x^3}$$
My book says that one can substitute $(r,s) = (x^2y,\ln{|x|})$ to get
$$\frac{ds}{dr}=\frac{1}{r^2-1}$$
The right-hand side of the ODE is not my problem. The right-hand side reduces to
$$\exp{(-3s)}(r^2-2r-1)$$
I just don't know how to replace $dy/dx$ by an expression depending on $ds/dr$. I would be very thankful if someone could show me how that works step by step. 

You have $$\frac{dr}{dx}=2xy+x^2\frac{dy}{dx}$$ and $$\frac{ds}{dx}=\frac 1x$$
Therefore, $$\frac{dr}{ds}=2x^2y+x^3\frac{dy}{dx}=2r+e^{3s}\frac{dy}{dx}$$
So now you can make the substitution.
