Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Question

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple)

implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. 1, grid = [30, 30]);

for $H := x^3-xy^2+1$ and the graph is 'empty' so I decided to expand the scale to

             implicitplot(x^3-x*y^2+1 = 0, x = -4 .. 4, y = -4 .. 4)        

and we get the following picture

My question is: How can I predict (mathematically) the fact that in $x = -1 .. 1, y = -1 .. 1$ there is no curve?

Answer 1

Because if $|x|<1$ and $|y|<1$, then $$-1<-y^2\le x^2-y^2\le x^2<1\implies |x^3-xy^2|=|x||x^2-y^2|<1\cdot 1=1,$$ so it's not possible that $x^3-xy^2=-1$.