AP Calculus AB › Unit 7
A differential equation describes how a quantity changes. In this unit you will write equations from verbal descriptions, verify solutions, sketch and read slope fields, and solve separable equations with an initial condition. Exponential growth and decay models round it out. Free-response questions often ask for a tangent line from the equation, then a full particular solution. Points disappear quickly if you skip separating the variables or omit the constant of integration. Some students also assume every fraction integrates to a natural log, which is not always true.
These are taken from the Regular set, with the answer and explanation shown.
1. Which function is a solution of the differential equation y″ − 4y = 0?
Answer: C. For y = 5e^(−2x), y′ = −10e^(−2x) and y″ = 20e^(−2x). Then y″ − 4y = 20e^(−2x) − 20e^(−2x) = 0.
Common trap: y = sin(2x) has y″ = −4 sin(2x), so y″ − 4y = −8 sin(2x). It solves y″ + 4y = 0 instead.
2. What is the general solution of dy/dx = y² sin x?
Answer: D. Separate: y⁻² dy = sin x dx. Integrate: −1/y = −cos x + C. Multiply by −1 and flip: y = 1/(cos x + C), after renaming −C as C.
Common trap: y = 1/(C − cos x) loses one minus sign. Its derivative is −sin x/(C − cos x)², which is −y² sin x.
3. A city's population P satisfies dP/dt = kP, where t is in years. P(0) = 2,000 and P(4) = 2,600. To the nearest whole number, what is P(10)?
Answer: A. P(t) = 2000e^(kt) with e^(4k) = 1.3. Then P(10) = 2000(1.3)^(10/4) = 2000(1.3)^2.5 ≈ 3,854.
Common trap: 3,500 assumes the population gains the same 150 people each year. Under dP/dt = kP, growth speeds up as P grows.