Inside This Unit: The Full Breakdown
Differential equations are equations involving derivatives. This unit covers solving separable differential equations, understanding slope fields as visual representations, and modeling exponential growth and decay.
Why it matters
Differential equations appear on the AP exam as both multiple-choice and free-response questions. Slope fields test conceptual understanding, while separable equations test procedural skill. Exponential growth and decay models are a common application context.
Key concepts
- A differential equation relates a function to its derivatives. A solution is a function that satisfies the equation.
- Slope fields visualize solutions by plotting short line segments with slope equal to dy/dx at each point.
- Separable equations can be written as g(y) dy = f(x) dx and solved by integrating both sides.
- Exponential growth/decay dy/dt = ky has the solution y = Ce^(kt), where k > 0 is growth and k < 0 is decay.
Slope Fields
A slope field is a graphical representation of a differential equation dy/dx = f(x, y). At each point in the plane, you draw a short segment with the slope given by f(x, y). Solution curves follow these segments like paths through the field. The AP exam asks you to sketch slope fields, match them to differential equations, and sketch particular solution curves through given initial conditions. Look for patterns: if dy/dx depends only on y, all segments in a horizontal row have the same slope.
Separable Differential Equations
A separable equation can be rearranged so all y-terms are on one side and all x-terms on the other: g(y) dy = f(x) dx. Integrate both sides, add the constant of integration, and solve for y if possible. Use the initial condition to find the particular value of C. Common mistakes include forgetting absolute values inside logarithms and neglecting to check that the solution satisfies the original equation. The AP free-response section regularly includes a separable equation requiring this procedure.
Exponential Models
The differential equation dy/dt = ky models exponential growth (k > 0) or decay (k < 0). Its general solution is y = Ce^(kt), where C is the initial value y(0). Radioactive decay, continuously compounded interest, and population growth under unlimited resources all follow this model. On the AP exam, you may be asked to set up the differential equation from a verbal description, solve it, or use it to predict future values. Always identify whether the context calls for growth or decay based on the sign of k.
AP exam tip
When solving a separable differential equation on the free response, show every step: separate variables, integrate both sides with + C on one side, apply the initial condition, then solve for y. Skipping the constant of integration is a common error that costs points.
Connections to other units
- Unit 5: Solving differential equations requires antidifferentiation skills from the integration unit.
- Unit 3: Differential equations model the same rate-of-change contexts seen in contextual applications.
- Unit 8: Accumulation and net change problems sometimes begin as differential equations.
