Review / Part I — Explicit & Qualitative Methods / Ch 1

Slope (Direction) Fields

Ch 1 — Part I — Explicit & Qualitative Methods

A direction (slope) field visualizes a first-order ODE \(y' = f(t, y)\) by drawing a short segment of slope \(f(t,y)\) at each point \((t,y)\) of the plane. Solutions are curves that are tangent to those segments.

Reading a slope field

Along the curve \(f(t,y) = 0\), solutions have horizontal tangent — these are called nullclines. Above a nullcline where \(f > 0\), solutions increase; below where \(f < 0\), they decrease.

Autonomous simplification

If \(f\) depends only on \(y\) (autonomous), the slope field is translation-invariant in \(t\): shifting a solution horizontally gives another solution.

Limitations

Slope fields are qualitative. They show you attractors, repellers, and rough shape, but not exact formulas. Numerical methods (Euler, RK4) trace solutions through the field accurately.

t y y=1
Slope field for \(y' = y(1-y)\) (logistic).

Practice Problems

Problem 1easy
Sketch where the slope field for \(y' = t - y\) has horizontal segments.
Hint
Set \(f(t,y) = 0\).
Solution

\(t - y = 0\) ⇒ the nullcline is the line \(y = t\). Along this line slopes are zero.

Answer: Along \(y = t\).

Problem 2easy
For the autonomous ODE \(y' = y^2 - 1\), describe the slope field along \(y = 1\) and along \(y = 2\).
Hint
Evaluate \(f(y)\).
Solution

At \(y = 1\), \(y' = 0\) (horizontal). At \(y = 2\), \(y' = 3\) (steep positive). The field is the same for all \(t\).

Answer: \(y=1\): horizontal; \(y=2\): slope 3.

Problem 3medium
True or false: two distinct solution curves of \(y' = f(t, y)\), where \(f\) and \(\partial f/\partial y\) are continuous, can cross.
Hint
Picard's uniqueness theorem.
Solution

Crossing would give two solutions with the same initial condition at the crossing point, contradicting uniqueness.

Answer: False.