riaology

Lesson Progress

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Big idea

A slope field is a behavior map.
At each point

(t,y)

, the DE tells you the slope

\frac{dy}{dt}

. If you can read the map, you can predict what the solutions will do before you even solve.

INTUITION

What a slope field shows (in one sentence)

Think: tiny arrows

Each little segment tells you how the solution curve must tilt at that point.

If the slope is positive, curves tilt up as time increases. If the slope is negative, curves tilt down. If the slope is 0, curves are flat there.

Quick check

choose one

If a DE has

\frac{dy}{dt}

positive everywhere in the region, what will solution curves do as t increases?

PLAY

Slope field plotter

Choose an equation

Try a few and look for equilibria and regions where slopes change sign.

Slope field plotter

Click anywhere on the plot to sample the slope at that point.

\frac{dy}{dt}=y

View window

t mint maxy miny max

Tip: Keep tMax > tMin and yMax > yMin. If the plot looks weird, reset to -3..3.

AxesNullclines (f=0)

Style + sampling

Density17×17

Segment length14px

Slope clamp (visual)±5

This only affects drawing.

Click the plot to sample a slope.

Exponential growth

Slopes get steeper as y increases.

In this preset: $\frac{dy}{dt}=y$

Above the x-axis, slopes are positive → solutions rise.
Below the x-axis, slopes are negative → solutions fall.
y = 0 is an equilibrium (flat slope).

Research move

Predict before solving

Without solving, ask:

Where are the slopes zero?

Above or below those, are slopes mostly positive or negative?

If you start near an equilibrium, do solutions move toward it or away?

PRACTICE

Practice: read the map

Practice progress

1) Reading equilibria

keyword check

In a slope field, what do flat little segments usually tell you?

Matched concepts: 0/2

Quick check

choose one

For

\frac{dy}{dt}=y(1-y)

, what happens to solutions that start between 0 and 1?

3) Autonomous equations

keyword check

Why do autonomous equations $\frac{dy}{dt}=f(y)$ often produce vertical stripe patterns in slope fields?

Matched concepts: 0/2

Quick check

choose one

If slopes are negative below a certain curve and positive above it, what does that separating curve usually represent?

WHAT'S NEXT

From slope fields → phase lines (autonomous DEs)

Next, we’ll specialize to autonomous equations $\frac{dy}{dt}=f(y)$ . Then the slope field has vertical stripe patterns, and you can summarize behavior with a phase line.

← Back to roadmap Next: Separable Equations →

Slope Fields: Behavior Maps

What a slope field shows (in one sentence)

Slope field plotter

Practice: read the map

From slope fields → phase lines (autonomous DEs)