The level surfaces of the equation $f(x,y)=y(x^2-y^2)$

Question

So as the title suggests that I have to find the level curves of the equation $f(x,y)=y(x^2-y^2)$ for f(x,y)= 1,0,-1. I trivially found that for 0 the equation simply becomes $x^2=y^2$ which is a cross. However, for 1 and -1 I can't wrap my head around it. I see that we define the equation as $Y(x^2-y^2)=C $ and we see that there is the term $x^2-y^2$ that can be converted to the hyperboloid. The problem with this equation is the Y outside of the brackets, due to this I have no clue how to convert it to the equation. $x^2/C+Y^2/C$ If somebody could provide me with help, it would greatly be appreciated!

Answer 1

The level curves are cubics $x^2y-y^3=c$

I draw some of them in the picture below.

Hope this helps

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