Slope Fields

Unit 7 · Differential Equations

Slope Fields and Differential Equations

A slope field (direction field) visualizes a differential equation dy/dx = f(x,y) by drawing short segments with slope f(x,y) at each grid point. Particular solutions follow the flow of these segments. Equilibrium solutions are constant functions y = k where f(x,k) = 0 for all x.

Key Ideas

Slope at (x, y)

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

General solution: family of curves; particular solution: one curve satisfying y(x₀) = y₀

Equilibrium: y = k where f(x, k) = 0 for all x

AP Tip: To match a slope field to its DE: look for where segments are horizontal (dy/dx = 0), whether slope depends on x alone, y alone, or both.

Type 1

Verifying Solutions and Finding Particular Solutions

Substitute y and dy/dx into the DE to verify. For particular solutions, apply the initial condition to the general solution to find C.

Example 1

Verify that

y = Ce^{3x}

satisfies

dy/dx = 3y

. Then find the particular solution with

y(0) = 2

Example 2

The general solution of dy/dx = 4x is y = 2x² + C. Find the particular solution with y(1) = 5.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 2

Reading and Computing Slopes

Evaluate dy/dx = f(x,y) at specific points, identify zero-slope isoclines, and classify equilibrium solutions.

Example 3

For dy/dx = x − y, compute slopes at (0,0), (1,0), (0,1), (1,1). On which line does dy/dx = 0?

Example 4

For dy/dx = 2 − y, find the equilibrium solution and classify the behavior at y(0) = 0.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 3

Matching Slope Fields to Differential Equations

Identify which DE corresponds to a described or drawn slope field by checking zero-slope conditions, sign patterns, and whether slope depends on x, y, or both.

Example 5

A slope field has horizontal segments along y = x, positive slopes above y = x, and negative slopes below. Which DE matches: (A) dy/dx = y−x, (B) dy/dx = x−y, (C) dy/dx = xy?

Practice more of this type— AI-generated · always-new problems

Generate Problems →

← Improper Integrals Separation of Variables →