Slope Fields Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Slope Fields.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A slope field draws the slope $\frac{dy}{dx}=f(x,y)$ as a short segment at each point, so solutions are curves that follow the arrows.

Common stuck point: The procedure for slope fields is the easy part; the trap is using $f(x,y)$ as a height instead of a slope. Asking "Am I representing a first-order DE as a grid of segments whose slopes the DE assigns at each point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I representing a first-order DE as a grid of segments whose slopes the DE assigns at each point?

Worked Examples

Example 1

easy

For

dy/dx = y

, compute slopes at

(0,0)

(0,1)

(1,1)

(1,-1)

. Describe the solution family.

Answer

Slopes:

0, 1, 1, -1

. Solutions:

y = Ce^x

First step

Slope

= y

at each point.

Full solution

2
$(0,0)$ : 0; $(0,1)$ : 1; $(1,1)$ : 1; $(1,-1)$ : $-1$ .
3
Solutions are $y = Ce^x$ : exponential curves.

When slope depends only on

y

(autonomous), the field has horizontal translation symmetry.

Example 2

medium

For

dy/dx = x-y

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

Example 3

medium

For

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

, find the isocline of slope

1

Example 4

medium

The DE

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

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

Example 5

medium

Use Euler's method with

h = 0.1

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

y(0) = 1

, to estimate

y(0.2)

Example 6

hard

For

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

, prove that no solution curve can have a local maximum in the upper half-plane

y > 0

Example 7

hard

Compare the slope fields of

\frac{dy}{dx} = y

and

\frac{dy}{dx} = -y

. Describe how their solutions differ.

Example 8

hard

At which points does the slope field of

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

fail to be defined?

Example 9

hard

For

\frac{dy}{dx} = 2xy

, find the solution through

(0, 3)

Example 10

challenge

For

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

, find a particular solution that grows like a polynomial as

x \to \infty

, and describe how every other solution behaves.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

For

dy/dx = x^2

, compute slopes at

(0,0)

(1,0)

(-1,3)

(2,5)

Example 2

medium

A slope field shows solutions curving toward the

x

-axis from both sides. What can you infer about

f(x,y)

and long-term behavior?

Example 3

easy

For

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

, what slope is drawn at

(2,3)

Example 4

easy

For

\frac{dy}{dx}=2

, describe the slope field.

Example 5

easy

For

\frac{dy}{dx}=x

, what is the slope along the

y

-axis (

x=0

Example 6

easy

For

\frac{dy}{dx}=y

, where are the slopes zero?

Example 7

easy

For

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

, what slope is drawn at the origin?

Example 8

easy

Can two distinct solution curves cross where

f(x,y)

is continuous?

Example 9

easy

How does a slope field differ from a vector field?

Example 10

easy

For

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

, what slope is drawn at

(1,-1)

Example 11

medium

For

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

, find the isocline where the slope is

1

Example 12

medium

For

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

, what is the slope at

(3,4)

Example 13

medium

For

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

, identify the equilibrium solutions.

Example 14

medium

For

\frac{dy}{dx}=y

, are slopes positive or negative above the

x

-axis?

Example 15

medium

Which DE has a slope field with horizontal segments exactly on the

x

-axis:

y'=x

y'=y

Example 16

medium

For

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

, where is the slope equal to

0

Example 17

medium

For

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

, compare slopes at

(0,1)

and

(1,0)

Example 18

medium

A solution passes through

(0,1)

for

\frac{dy}{dx}=y

. Is it increasing there?

Example 19

medium

For

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

, find the slope at

(2,1)

Example 20

challenge

For

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

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

x=1

Example 21

challenge

Why do two apparent solution curves crossing in a slope field indicate a problem?

Example 22

challenge

For

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

, show the slope field is consistent with circular solutions.

Example 23

easy

For

\frac{dy}{dx} = 2x

, what slope is drawn at

(3, 7)

Example 24

easy

For

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

, find the equilibrium (zero-slope) line.

Example 25

easy

For

\frac{dy}{dx} = x y

, what slope is drawn at

(2, -3)

Example 26

easy

For

\frac{dy}{dx} = \sin x

, at which

x

-values is the slope zero?

Example 27

easy

For

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

, find the slope at

(0, -2)

Example 28

medium

For

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

, find the isocline of slope

0

Example 29

medium

For

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

(

y \neq 0

), what family of curves are the solutions?

Example 30

medium

For

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

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

Example 31

medium

Match the DE

\frac{dy}{dx} = -y

to its solution family.

Example 32

medium

For

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

, what slope is drawn at

(4, 2)

Example 33

hard

For

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

(with

x > 0

), find the family of solution curves passing through

(1, 2)

Example 34

hard

For

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

, identify the equilibria and classify their stability.

Example 35

hard

For

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

, what can you say about the long-term behavior of any solution as

x \to \infty

Example 36

challenge

Sketch-style: for

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

, find every isocline of slope

0

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Related Concepts

Introduction to Differential Equations Derivative Separation of Variables

Background Knowledge

These ideas may be useful before you work through the harder examples.

differential equations introderivative