Slope Fields
This type of diagram uses small line segments to show the slope of dy/dx.
What is a slope field?
A solution that approaches a horizontal line as x → ∞ is said to be this.
What is approaching an equilibrium?
dy/dx = 0 produces this type of solution.
What is an equilibrium solution?
This determines the unique solution curve from a family of solutions.
What is the initial condition?
Solutions move toward this type of equilibrium as t → ∞.
What is a stable equilibrium?
If dy/dx = y(1 − y), what happens to solutions when 0 < y < 1?
What is they increase toward y = 1?
Equilibrium solutions are always this type of function.
What is constant?
Solve: dy/dx = 2x, y(0) = 3.
What is y = x² + 3?
True or False: A slope field can represent the exact solution to a differential equation.
What is false? (It shows qualitative behavior.)
Describe the behavior of solutions to dy/dx = −y.
What is exponential decay toward 0?
For dy/dx = y(1 − y), identify all equilibrium solutions.
What is y = 0 and y = 1?
dy/dx = y, y(0) = 2. What is the general solution?
What is y = Ce^x? (C = 2)
dy/dx = x − y produces what kind of general slope pattern near the line y = x?
What is horizontal?
A solution that moves away from an equilibrium is said to be this.
What is unstable?
Describe the stability of y = 0 for dy/dx = y(1 − y).
What is unstable?
Describe the long-term behavior of y = Ce^(−x), with C > 0.
What is exponential decay to 0?
You see a slope field with horizontal segments along y = 3. What is dy/dx likely equal to?
You see a slope field with horizontal segments along y = 3. What is dy/dx likely equal to?
If y(0) = 2 and dy/dx = y(y − 3), describe the long-term behavior of the solution.
What is it increases without bound (goes to ∞)?
For dy/dx = y² − 4, classify the equilibria and their stability.
What is y = −2 (unstable), y = 2 (unstable)?
Given the slope field for dy/dx = y(1 − y), sketch the solution through y(0) = 0.5.
What is a curve that increases toward y = 1?