Multiple Choice Questions
Calculus: Differential Equations and Applications
Topic: Calculus: Differential Equations and Applications
Grade: 12
Question 1:
Which of the following is a solution to the differential equation dy/dx = 2x?
a) y = x^2 + 3
b) y = x^2 – 2x
c) y = x^2 + 2
d) y = x^2 + 1
Answer: b) y = x^2 – 2x
Explanation: The given differential equation is dy/dx = 2x. To solve this equation, we integrate both sides with respect to x. The integral of 2x with respect to x is x^2, and the integral of dy/dx is y. Therefore, the general solution to the differential equation is y = x^2 + C, where C is a constant. In this case, the constant is -2 since we have y = x^2 – 2x. This can be verified by taking the derivative of y = x^2 – 2x and confirming that it equals 2x.
Example 1: For x = 3, the value of y in the solution y = x^2 – 2x is y = 3^2 – 2(3) = 9 – 6 = 3.
Example 2: For x = -2, the value of y in the solution y = x^2 – 2x is y = (-2)^2 – 2(-2) = 4 + 4 = 8.
Question 2:
Which of the following differential equations represents exponential growth?
a) dy/dx = x
b) dy/dx = y
c) dy/dx = -x
d) dy/dx = -y
Answer: b) dy/dx = y
Explanation: Exponential growth is represented by the differential equation dy/dx = ky, where k is a constant. This means that the rate of change of y is proportional to its current value. In this case, the differential equation dy/dx = y satisfies this condition.
Example 1: For y = 2, the value of dy/dx in the solution dy/dx = y is dy/dx = 2.
Example 2: For y = 5, the value of dy/dx in the solution dy/dx = y is dy/dx = 5.
Question 3:
The solution to the differential equation dy/dx = e^x is:
a) y = e^x + C
b) y = e^x – C
c) y = e^x * C
d) y = e^x / C
Answer: a) y = e^x + C
Explanation: The given differential equation is dy/dx = e^x. To solve this equation, we integrate both sides with respect to x. The integral of e^x with respect to x is e^x, and the integral of dy/dx is y. Therefore, the general solution to the differential equation is y = e^x + C, where C is a constant. This can be verified by taking the derivative of y = e^x + C and confirming that it equals e^x.
Example 1: For x = 2, the value of y in the solution y = e^x + C is y = e^2 + C.
Example 2: For x = -1, the value of y in the solution y = e^x + C is y = e^-1 + C.
Question 4:
Which of the following is a particular solution to the differential equation dy/dx = 3x^2?
a) y = x^3 + C
b) y = x^3 – C
c) y = x^3 / 3 + C
d) y = x^3 / 3 – C
Answer: c) y = x^3 / 3 + C
Explanation: The given differential equation is dy/dx = 3x^2. To find a particular solution, we can integrate both sides with respect to x. The integral of 3x^2 with respect to x is x^3, and the integral of dy/dx is y. Therefore, the particular solution to the differential equation is y = x^3 / 3 + C, where C is a constant. This can be verified by taking the derivative of y = x^3 / 3 + C and confirming that it equals 3x^2.
Example 1: For x = 2, the value of y in the particular solution y = x^3 / 3 + C is y = 2^3 / 3 + C.
Example 2: For x = -1, the value of y in the particular solution y = x^3 / 3 + C is y = (-1)^3 / 3 + C.