help to parametrize $y^2 = x^3 -x^2$

Question

I appreciate if someone could help me to parametrize this equation $y^2 = x^3 -x^2$.

Thanks in advance. I used maple to find the solution as $(x,y) = ((t^2-1),(t(t^2-1))$

Answer 1

What you do is plug the parameterization into the equation. You want to find that if you plug $t^2-1$ in for $x$ you get the given expression for $y$. Unfortunately, it fails. So $$ \begin {align} x^3-x^2 &=(t^2-1)^3-(t^2-1)^2\\ &=(t^2-1)^2(t^2-1-1)\\ &=(t^2-1)^2(t^2-2)\\&=y^2-2x^2 \end {align}$$ Could it be your equation is supposed to be $x^3+x^2=y^2$?