Example 3 — The Surface \( f(x,y) = (1-xy)^3 \)

The surface

The surface \( z = (1-xy)^3 \) (heights clipped for display) with the isoclines \( xy=1 \) and \( xy=2 \)

Given the ODE

\( y' = (1 - xy)^3, \)

we sketch the corresponding surface

\( f(x,y) = (1 - xy)^3. \)

On each isocline of height \( C \),

\( (1 - xy)^3 = C, \)

we draw vectors whose slope is \( C \).

Vectors on each isocline

For \( C = -8 \): on the curve \( (1-xy)^3 = -8 \) we get \( xy = 3 \), and we draw on the curve vectors with direction \( \theta = \arctan(-8) \approx -83^\circ \).

For \( C = -1 \): on the curve \( (1-xy)^3 = -1 \) we get \( xy = 2 \), and we draw on the curve vectors with direction \( \theta = \arctan(-1) = -45^\circ \).

For \( C = 0 \): on the curve \( (1-xy)^3 = 0 \) we get \( xy = 1 \), and we draw on the curve vectors with direction \( \theta = \arctan(0) = 0^\circ \).

For \( C = 1 \): on the curve \( (1-xy)^3 = 1 \) we get \( xy = 0 \), and the field vector makes the angle \( \theta = \arctan(1) = 45^\circ \).

Hyperbola isoclines \( xy = 3, 2, 1 \) and the axes \( xy = 0 \), each carrying vectors of one constant direction

The dense direction field and five solutions

The dense direction field of \( y' = (1-xy)^3 \) and five solutions

Continuing this way we obtain the dense direction field. Note that the vectors of the same isocline all point in the same direction — that direction is \( \arctan C \), where \( C \) is the height of the isocline on the surface \( f(x,y) = (1-xy)^3 \).

The solutions of the equation must be tangent to the dense direction field. The panel shows five solutions of the equation for the initial conditions:

\( (-2,\, 1.5), \quad (0,\, 0), \quad (2,\, -1.2), \quad (1,\, 1.5), \quad (0.5,\, 2.5). \)
© Shir Sivroni