Practice Slope Fields in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A graphical representation of a first-order DE $\frac{dy}{dx} = f(x, y)$ . At each point $(x, y)$ in the plane, draw a short line segment with slope $f(x, y)$ . The resulting pattern of segments shows the direction solutions must follow.

Imagine a field with tiny arrows showing which way a river flows at each point. A slope field is the same idea: the DE tells you the slope (direction) at every point, and solution curves are paths that follow these directions everywhere. Drop a 'particle' anywhere and follow the arrows—that's a solution.

Showing a random 20 of 50 problems.

Example 1

medium

For

\frac{dy}{dx}=x^2-y

, find the slope at

(2,1)

Example 2

hard

Compare the slope fields of

\frac{dy}{dx} = y

and

\frac{dy}{dx} = -y

. Describe how their solutions differ.

Example 3

easy

For

\frac{dy}{dx}=x+y

, what slope is drawn at the origin?

Example 4

easy

For

\frac{dy}{dx} = y^2

, find the slope at

(0, -2)

Example 5

challenge

For

\frac{dy}{dx}=-\frac{x}{y}

, show the slope field is consistent with circular solutions.

Example 6

hard

For

\frac{dy}{dx} = (y - 2)(y + 1)

, identify the equilibria and classify their stability.

Example 7

medium

For

\frac{dy}{dx} = y - x

, what slope is drawn at

(4, 2)

Example 8

challenge

For

\frac{dy}{dx}=x+y

, sketch how the slope of solutions changes moving up a vertical line

x=1

Example 9

medium

For

\frac{dy}{dx}=x+y

, compare slopes at

(0,1)

and

(1,0)

Example 10

medium

For

\frac{dy}{dx} = -\frac{x}{y}

(

y \neq 0

), what family of curves are the solutions?

Example 11

easy

For

\frac{dy}{dx}=2

, describe the slope field.

Example 12

easy

True or false: solution curves of

\frac{dy}{dx} = f(x, y)

must be tangent to the slope field segments.

Example 13

medium

For

\frac{dy}{dx} = x - y

, find the isocline of slope

1

Example 14

medium

The DE

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

models logistic growth. Find all equilibrium lines and classify each as stable or unstable.

Example 15

easy

For

\frac{dy}{dx}=x+y

, what slope is drawn at

(1,-1)

Example 16

medium

For

dy/dx = x-y

, find the zero-slope isocline and describe long-term behavior of solutions.

Example 17

challenge

Sketch-style: for

\frac{dy}{dx} = \sin(x + y)

, find every isocline of slope

0

Example 18

easy

For

dy/dx = x^2

, compute slopes at

(0,0)

(1,0)

(-1,3)

(2,5)

Example 19

easy

For

\frac{dy}{dx}=x

, what is the slope along the

y

-axis (

x=0

Example 20

medium

For

\frac{dy}{dx} = 1 + y^2

, what is the smallest slope drawn anywhere in the field?

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Related Concepts

Introduction to Differential Equations Derivative Separation of Variables