Differential Equations Practice Questions
AP (Advanced Placement) · AP Calculus BC · 149 free MCQs with instant results and detailed explanations.
149
Total
39
Easy
79
Medium
31
Hard
Start Practicing Differential Equations
Take a timed quiz or customize your practice session
Sample Questions from Differential Equations
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the general solution to the differential equation dy/dx = 6x?
A.
3x^2 + C
B.
6x + C
C.
x^6 + C
D.
12x + C
Show Answer & Explanation
Correct Answer: A
The integral of 6x with respect to x is 3x^2, hence the general solution is 3x^2 + C, where C is the constant of integration.
Q2
Easy
If y = e^(3x) is a solution to the differential equation dy/dx = ky, what is the value of k?
A.
3
B.
e^3
C.
0
D.
1
Show Answer & Explanation
Correct Answer: A
To find k, we differentiate y with respect to x. dy/dx = 3e^(3x). Since dy/dx = ky, we equate 3e^(3x) = k(e^(3x)), which gives k = 3.
Q3
Easy
Which of the following differential equations is separable?
A.
dy/dx = (2y)/(x^2)
B.
dy/dx = y^2 + x^2
C.
dy/dx = y + sin(x)
D.
dy/dx = x/y
Show Answer & Explanation
Correct Answer: A
The equation dy/dx = (2y)/(x^2) can be rearranged to (1/y) dy = (2/x^2) dx, allowing us to separate the variables for integration.
Q4
Medium
What is the particular solution of the differential equation dy/dx = 4x^3 with the condition y(1) = 5?
A.
y = x^4 + 1
B.
y = x^4 + 4
C.
y = x^4 + 5
D.
y = x^4 + 2
Show Answer & Explanation
Correct Answer: B
To find the particular solution, we integrate dy/dx = 4x^3 to get y = x^4 + C. Applying the initial condition y(1) = 5 gives C = 4, resulting in the particular solution y = x^4 + 4.
Q5
Medium
For the differential equation y' = y(2 - y), what is the equilibrium solution?
A.
y = 2
B.
y = 0
C.
y = 1
D.
y = 3
Show Answer & Explanation
Correct Answer: A
Equilibrium solutions occur when y' = 0. Setting y(2 - y) = 0 gives y = 0 or y = 2. Therefore, the equilibrium solution in this context is y = 2.
Q6
Medium
Which of the following represents a particular solution to the differential equation dy/dx = 2x with the initial condition y(0) = 5?
A.
y = x^2 + 5
B.
y = 2x + 5
C.
y = x^2 + 2
D.
y = 2x^2 + 5
Show Answer & Explanation
Correct Answer: A
To find the particular solution, we integrate dy/dx = 2x, yielding y = x^2 + C. Using the initial condition y(0) = 5, we find C = 5, leading to the solution y = x^2 + 5.
Q7
Medium
If the function y(t) satisfies the differential equation dy/dt = -4y + 12, what is the steady state solution?
A.
3
B.
2
C.
4
D.
6
Show Answer & Explanation
Correct Answer: A
The steady state solution occurs when dy/dt = 0. Setting -4y + 12 = 0 gives y = 3. Therefore, the steady state solution is 3.
Q8
Hard
Given the differential equation dy/dx = 3y + 2e^(2x), what is the general solution?
A.
y = C e^(3x) - e^(2x)
B.
y = C e^(3x) + e^(2x)
C.
y = C e^(2x) + e^(3x)
D.
y = C e^(2x) - e^(3x)
Show Answer & Explanation
Correct Answer: A
To solve the differential equation, we can use the integrating factor method. The integrating factor is e^(3x), leading to the general solution y = C e^(3x) - e^(2x).
Q9
Hard
Consider the differential equation dy/dx = y^2 - 4y + 3. What are the equilibrium solutions?
A.
y = 1, y = 3
B.
y = 2
C.
y = 0, y = 4
D.
y = -1, y = 5
Show Answer & Explanation
Correct Answer: A
To find the equilibrium solutions, set dy/dx = 0. This gives y^2 - 4y + 3 = 0, which factors to (y - 1)(y - 3) = 0. Hence, the equilibrium solutions are y = 1 and y = 3.
Q10
Hard
Solve the initial value problem: dy/dx = 3y + 2, with y(0) = 1. What is the value of y when x = 1?
A.
e^3 - 1/3
B.
2e^3 - 1/3
C.
e^3 + 1/3
D.
e^3 + 1
Show Answer & Explanation
Correct Answer: A
To solve the differential equation dy/dx = 3y + 2, separate variables and integrate. The solution is y = (2/3)e^(3x) - (2/3). Using the initial condition y(0) = 1, we find the constant and evaluate y(1) to obtain e^3 - 1/3.
Showing 10 of 149 questions. Start a quiz to practice all questions with scoring and timer.
Practice All 149 Questions →Other AP (Advanced Placement) Subjects
Differential Equations — AP (Advanced Placement) AP Calculus BC Practice Questions Online
This page contains 149 practice MCQs for the chapter Differential Equations in AP (Advanced Placement) AP Calculus BC. The questions are organized by difficulty — 39 easy, 79 medium, 31 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.