AP Calc AB Differential Equations

Question 1

What is the particular solution to dy/dx = 2xy with initial condition y(0) = 3?

Accepted Answer

y = 3e^(x²)

Answer

Separating gives dy/y = 2x dx, so ln|y| = x² + C and y = Ke^(x²). Using y(0) = 3 gives K = 3.

Question 2

A slope field is drawn for dy/dx = x − y. At which of the following points is the slope segment horizontal?

Accepted Answer

(2, 2)

Answer

A horizontal segment has slope 0, so x − y = 0, which means x = y. Only (2, 2) satisfies this.

Question 3

Which function is the solution to dy/dx = −x/y with initial condition y(0) = 2?

Accepted Answer

y = √(4 − x²)

Answer

Separating gives y dy = −x dx, so y²/2 = −x²/2 + C and y² + x² = 4. Since y(0) = 2 is positive, y = √(4 − x²).

Question 4

If dy/dx = y² and y(0) = 1, what is y(1/2)?

Accepted Answer

2

Answer

Separating gives dy/y² = dx, so −1/y = x + C. With y(0) = 1, C = −1, so y = 1/(1 − x) and y(1/2) = 2.

Question 5

A function y = f(t) satisfies dy/dt = −3y with f(0) = 6. What is f(ln 2)?

Accepted Answer

3/4

Answer

The solution is y = 6e^(−3t). At t = ln 2, e^(−3 ln 2) = 2^(−3) = 1/8, so f(ln 2) = 6/8 = 3/4.

AP Calc AB Practice Test