AP Calculus AB Unit 7: Differential Equations
Differential Equations
Slope Fields, Separation of Variables, and Modeling
1. What is a Differential Equation?
A differential equation is an equation involving a function and its derivatives.
dy
Example: = 3x2 + 1
dx
• General Solution: Contains an arbitrary constant (+C). Represents a family of functions.
• Particular Solution: A specific solution found using an initial condition (e.g., y(0) = 5).
2. Slope Fields
dy
A graphical representation of a differential equation dx = f (x, y).
Interpreting Slope Fields
• At every point (x, y), the small line segment represents the slope of the tangent line.
• To sketch a solution curve: Start at the initial point and "follow the flow" of the tick marks.
dy
• If dx depends only on x, slopes are constant vertically (columns match).
dy
• If dx depends only on y, slopes are constant horizontally (rows match).
3. Separation of Variables
The primary algebraic method used in AB Calculus to solve differential equations.
1
Differential Equations
Slope Fields, Separation of Variables, and Modeling
1. What is a Differential Equation?
A differential equation is an equation involving a function and its derivatives.
dy
Example: = 3x2 + 1
dx
• General Solution: Contains an arbitrary constant (+C). Represents a family of functions.
• Particular Solution: A specific solution found using an initial condition (e.g., y(0) = 5).
2. Slope Fields
dy
A graphical representation of a differential equation dx = f (x, y).
Interpreting Slope Fields
• At every point (x, y), the small line segment represents the slope of the tangent line.
• To sketch a solution curve: Start at the initial point and "follow the flow" of the tick marks.
dy
• If dx depends only on x, slopes are constant vertically (columns match).
dy
• If dx depends only on y, slopes are constant horizontally (rows match).
3. Separation of Variables
The primary algebraic method used in AB Calculus to solve differential equations.
1
