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
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.
Verify that y = Ce^{3x} satisfies dy/dx = 3y. Then find the particular solution with y(0) = 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 · infinite problems
Generate Problems →Reading and Computing Slopes
Evaluate dy/dx = f(x,y) at specific points, identify zero-slope isoclines, and classify equilibrium solutions.
For dy/dx = x − y, compute slopes at (0,0), (1,0), (0,1), (1,1). On which line does dy/dx = 0?
For dy/dx = 2 − y, find the equilibrium solution and classify the behavior at y(0) = 0.
Practice more of this type— AI-generated · infinite problems
Generate Problems →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.
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 · infinite problems
Generate Problems →