Isoclines and Nullclines

Given an ODE of the form

\( y' = f(x,y), \)

the value \( f(x_0, y_0) \) is the slope of the tangent to the solution passing through the point \( (x_0, y_0) \). So to sketch the direction field of the ODE, we want to know at which points of the plane the same slope occurs.

Instead of computing the slope separately at every point, we can partition the plane into sets of points that share the same slope. For a constant \( C \), all the points satisfying

\( f(x,y) = C \)

are points where the tangent slope of the solution equals \( C \). Such a set of points is called an isocline.

Careful to distinguish

The isocline itself is not necessarily a solution of the ODE. It only marks where in the plane the solutions attain the same slope. On each isocline \( f(x,y) = C \) we draw direction segments with that same slope \( C \).

The slope of the direction segments is \( C \), and their inclination angle relative to the \( x \)-axis is

\( \theta = \arctan(C). \)

For example, if \( C = 1 \), the slope is \( 1 \), so the direction angle is \( \arctan(1) = 45^\circ \). If \( C = 2 \), we get a larger angle: \( \arctan(2) \approx 63^\circ \). In general, the larger \( C \) is, the steeper the direction segments.

Building the skeleton of the direction field

To get a good geometric picture of the ODE, one isocline is not enough — a single isocline gives information only about the points where the slope equals one particular value. Sketching several isoclines for different values of \( C \) produces a kind of skeleton of the direction field: on each isocline we mark direction segments of constant slope, and an overall picture emerges of how the solutions are expected to behave.

For example, for the ODE

\( y' = x^2 + y^2 \)

the slope function is \( f(x,y) = x^2 + y^2 \), so the isoclines are given by

\( x^2 + y^2 = C. \)

For \( C > 0 \) these are circles centred at the origin. On each such circle the direction segments have slope \( C \), hence inclination \( \arctan(C) \):

\( \begin{aligned} C=1 &\Rightarrow \theta = \arctan(1) = 45^\circ,\\ C=2 &\Rightarrow \theta = \arctan(2) \approx 63^\circ,\\ C=4 &\Rightarrow \theta = \arctan(4) \approx 76^\circ,\\ C=8 &\Rightarrow \theta = \arctan(8) \approx 83^\circ. \end{aligned} \)

Isoclines of \( y' = x^2+y^2 \): on each circle \( x^2+y^2=C \), all direction segments share the slope \( C \)

Thus the farther we move from the origin, the larger \( x^2 + y^2 \) becomes, and the steeper the direction segments. For \( C = 0 \) we get the single point \( (0,0) \) — the nullcline of the ODE, i.e. the locus where the slope is \( 0 \). For \( C < 0 \) there are no points satisfying \( x^2 + y^2 = C \).

From isoclines to the full direction field

The direction field of \( y' = x^2+y^2 \) assembling isocline by isocline, then two solution curves

Continuing this way over many values of \( C \) fills the plane with direction segments — the direction field of the ODE. The solutions of the equation must be tangent to the direction field at every point, so once the skeleton is in place, the behaviour of the solution curves can be read directly from the picture.