Unit 7
Differential Equations
6–12% AB | 6–9% BCCED Unit 7
Modeling & Verifying
Translating verbal descriptions into differential equations and verifying proposed solutions.
Key Definitions
Differential Equation
An equation involving a derivative. Example: dy/dx = 2xy.
General Solution
The family of all solutions, including an arbitrary constant C.
Particular Solution
A specific solution found by using an initial condition to determine C.
Modeling Differential Equations
Translate verbal statements into differential equations:
| Statement | Differential Equation |
|---|---|
| “Rate of change of y is proportional to y” | |
| “Rate of change of P is proportional to P and (L − P)” | |
| “The slope at any point equals twice the y-coordinate” |
Verifying Solutions
1. Differentiate the proposed solution
2. Substitute into the differential equation
3. Confirm both sides are equal
→ Tip
To verify: differentiate, plug in, check equality. If the initial condition is given, also verify the solution satisfies it.
AP® Calculus AB & BC · Unit 7 Overview · Mr. Brantley