Cubic Isoclines — \( f(x,y) = (1-xy)^{3} \)

The surface

For

\( y' = (1-xy)^{3} \)

the slope surface \( z=(1-xy)^{3} \) is far less obvious than a plane or a paraboloid — the rotation reveals its saddle-like shape.

The surface — The rotating surface \( z=(1-xy)^{3} \)

The isoclines

The isoclines — Hyperbola isoclines \( xy = 1-\sqrt[3]{C} \)

Setting \( (1-xy)^{3} = C \) gives \( xy = 1 - \sqrt[3]{C} \) — a family of hyperbolas (and the line pair \( xy=1 \) for \( C=0 \), the nullcline).

Isoclines on the field

Overlaying the hyperbolas on the direction field shows each one carrying segments of a single slope.

Isoclines on the field — Isoclines over the direction field

Solutions

Solutions — Solutions threading the field of \( y'=(1-xy)^{3} \)

Solution curves follow the field, steep where \( |1-xy| \) is large and flat along the nullcline \( xy=1 \).