Differential Equations
Q. Find the general solution of dy/dx = 3x^2. (2020)
-
A.
y = x^3 + C
-
B.
y = 3x^3 + C
-
C.
y = x^2 + C
-
D.
y = 3x + C
Solution
Integrating 3x^2 gives y = x^3 + C.
Correct Answer: A — y = x^3 + C
Learn More →
Q. Find the general solution of the equation y' = 3x^2y.
-
A.
y = Ce^(x^3)
-
B.
y = Ce^(3x^3)
-
C.
y = C/x^3
-
D.
y = Cx^3
Solution
This is a separable equation. Integrating gives y = Ce^(x^3).
Correct Answer: A — y = Ce^(x^3)
Learn More →
Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(x)
-
D.
y = 2e^(x)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer: A — y = 2e^(4x)
Learn More →
Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(2x)
-
D.
y = 2e^(x/4)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer: A — y = 2e^(4x)
Learn More →
Q. Find the solution of the differential equation dy/dx = y^2.
-
A.
y = 1/(C - x)
-
B.
y = C/(x - 1)
-
C.
y = Cx
-
D.
y = e^(x)
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
Learn More →
Q. Find the solution of the differential equation y' = 3y + 6.
-
A.
y = Ce^(3x) - 2
-
B.
y = Ce^(3x) + 2
-
C.
y = 2e^(3x)
-
D.
y = 3Ce^(x)
Solution
This is a linear first-order equation. The integrating factor is e^(3x). The solution is y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
Learn More →
Q. Find the solution of the equation dy/dx = y^2 - 1.
-
A.
y = tan(x + C)
-
B.
y = C/(1 - Cx)
-
C.
y = 1/(C - x)
-
D.
y = C/(x + 1)
Solution
This is a separable equation. The solution is y = tan(x + C).
Correct Answer: A — y = tan(x + C)
Learn More →
Q. Solve the differential equation dy/dx = 2x + 1.
-
A.
y = x^2 + x + C
-
B.
y = x^2 + 2x + C
-
C.
y = 2x^2 + x + C
-
D.
y = x^2 + C
Solution
Integrating both sides, we get y = ∫(2x + 1)dx = x^2 + x + C.
Correct Answer: A — y = x^2 + x + C
Learn More →
Q. Solve the differential equation dy/dx = 2y + 3. (2023)
-
A.
y = Ce^(2x) - 3/2
-
B.
y = Ce^(-2x) + 3/2
-
C.
y = 3e^(2x)
-
D.
y = 2e^(2x) + C
Solution
Using an integrating factor, we find the solution is y = Ce^(2x) - 3/2.
Correct Answer: A — y = Ce^(2x) - 3/2
Learn More →
Q. Solve the differential equation dy/dx = y/x. (2023)
-
A.
y = Cx
-
B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is a separable equation. Integrating gives y = Cx.
Correct Answer: A — y = Cx
Learn More →
Q. Solve the differential equation y' = 5 - 2y.
-
A.
y = 5/2 + Ce^(-2x)
-
B.
y = 5 + Ce^(-2x)
-
C.
y = 2 + Ce^(2x)
-
D.
y = 5/2 - Ce^(-2x)
Solution
This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).
Correct Answer: A — y = 5/2 + Ce^(-2x)
Learn More →
Q. Solve the differential equation y'' - 3y' + 2y = 0.
-
A.
y = C1e^(2x) + C2e^(x)
-
B.
y = C1e^(x) + C2e^(2x)
-
C.
y = C1e^(-x) + C2e^(-2x)
-
D.
y = C1e^(3x) + C2e^(x)
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).
Correct Answer: B — y = C1e^(x) + C2e^(2x)
Learn More →
Q. Solve the first-order differential equation dy/dx = y/x.
-
A.
y = Cx
-
B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is a separable equation. Separating variables and integrating gives y = Cx.
Correct Answer: A — y = Cx
Learn More →
Q. Solve the first-order linear differential equation dy/dx + y/x = 1.
-
A.
y = x + C/x
-
B.
y = Cx - x
-
C.
y = Cx + x
-
D.
y = C/x + x
Solution
Using the integrating factor e^(∫(1/x)dx) = x, we solve to get y = x + C/x.
Correct Answer: A — y = x + C/x
Learn More →
Q. What is the general solution of the equation y' = 4y + 3?
-
A.
y = Ce^(4x) - 3/4
-
B.
y = Ce^(4x) + 3/4
-
C.
y = 3e^(4x)
-
D.
y = Ce^(3x) + 4
Solution
The integrating factor is e^(-4x). The solution is y = Ce^(4x) + 3/4.
Correct Answer: B — y = Ce^(4x) + 3/4
Learn More →
Q. What is the general solution of the equation y'' - 3y' + 2y = 0?
-
A.
y = C1 e^(x) + C2 e^(2x)
-
B.
y = C1 e^(2x) + C2 e^(x)
-
C.
y = C1 e^(3x) + C2 e^(0)
-
D.
y = C1 e^(0) + C2 e^(3x)
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. Thus, the general solution is y = C1 e^(2x) + C2 e^(x).
Correct Answer: B — y = C1 e^(2x) + C2 e^(x)
Learn More →
Q. What is the integrating factor for the equation dy/dx + (1/x)y = 2?
-
A.
x
-
B.
e^(ln(x))
-
C.
e^(ln(x^2))
-
D.
1/x
Solution
The integrating factor is e^(∫(1/x)dx) = e^(ln(x)) = x.
Correct Answer: C — e^(ln(x^2))
Learn More →
Q. What is the integrating factor for the equation dy/dx + 2y = 3?
-
A.
e^(2x)
-
B.
e^(-2x)
-
C.
e^(3x)
-
D.
e^(-3x)
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
Learn More →
Q. What is the integrating factor for the equation dy/dx + 2y = 6?
-
A.
e^(2x)
-
B.
e^(-2x)
-
C.
e^(6x)
-
D.
e^(-6x)
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
Learn More →
Q. What is the particular solution of dy/dx = 4y with the initial condition y(0) = 2?
-
A.
y = 2e^(4x)
-
B.
y = e^(4x)
-
C.
y = 4e^(4x)
-
D.
y = 2e^(x)
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer: A — y = 2e^(4x)
Learn More →
Q. What is the solution of the differential equation y' = 2y + 3?
-
A.
y = Ce^(2x) - 3/2
-
B.
y = Ce^(2x) + 3/2
-
C.
y = 3e^(2x)
-
D.
y = 2e^(x) + C
Solution
The integrating factor is e^(-2x). Solving gives y = Ce^(2x) + 3/2.
Correct Answer: B — y = Ce^(2x) + 3/2
Learn More →
Q. What is the solution of the equation dy/dx = 3x^2?
-
A.
y = x^3 + C
-
B.
y = 3x^3 + C
-
C.
y = x^2 + C
-
D.
y = 3x^2 + C
Solution
Integrating both sides gives y = ∫3x^2 dx = x^3 + C.
Correct Answer: A — y = x^3 + C
Learn More →
Q. What is the solution of the equation dy/dx = 4y + 2? (2021)
-
A.
y = Ce^(4x) - 1/2
-
B.
y = Ce^(-4x) + 1/2
-
C.
y = 2e^(4x) + C
-
D.
y = 4e^(4x) + C
Solution
Using an integrating factor, the solution is y = Ce^(4x) - 1/2.
Correct Answer: A — y = Ce^(4x) - 1/2
Learn More →
Q. What is the solution of the equation dy/dx = 6 - 2y? (2021)
-
A.
y = 3 - Ce^(-2x)
-
B.
y = 3 + Ce^(-2x)
-
C.
y = 2 - Ce^(2x)
-
D.
y = 6 - Ce^(2x)
Solution
Rearranging gives dy/(6 - 2y) = dx. Integrating both sides leads to y = 3 - Ce^(-2x).
Correct Answer: A — y = 3 - Ce^(-2x)
Learn More →
Q. What is the solution to the differential equation dy/dx = -y/x?
-
A.
y = Cx
-
B.
y = C/x
-
C.
y = Cx^2
-
D.
y = Cx^(-1)
Solution
This is a separable equation. Separating variables and integrating gives y = C/x.
Correct Answer: B — y = C/x
Learn More →
Q. What is the solution to the equation dy/dx = -5y?
-
A.
y = Ce^(-5x)
-
B.
y = -5Ce^x
-
C.
y = Ce^(5x)
-
D.
y = 5Ce^(-x)
Solution
This is a separable differential equation. The solution is y = Ce^(-5x), where C is a constant.
Correct Answer: A — y = Ce^(-5x)
Learn More →
Q. What is the solution to the equation dy/dx = y^2? (2022)
-
A.
y = 1/(C - x)
-
B.
y = C/(x - 1)
-
C.
y = Cx^2
-
D.
y = ln(Cx)
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
Learn More →
Q. What is the solution to the equation y' + 2y = 0?
-
A.
y = Ce^(-2x)
-
B.
y = Ce^(2x)
-
C.
y = 2Ce^x
-
D.
y = Ce^x
Solution
This is a separable equation. The solution is y = Ce^(-2x).
Correct Answer: A — y = Ce^(-2x)
Learn More →
Q. What is the solution to the equation y' + 3y = 0?
-
A.
y = Ce^(-3x)
-
B.
y = Ce^(3x)
-
C.
y = 3Ce^(-x)
-
D.
y = Ce^(-x/3)
Solution
This is a first-order linear differential equation. The solution is y = Ce^(-3x).
Correct Answer: A — y = Ce^(-3x)
Learn More →
Q. What is the solution to the equation y' = 3y + 6?
-
A.
y = Ce^(3x) - 2
-
B.
y = Ce^(3x) + 2
-
C.
y = 2e^(3x)
-
D.
y = 3Ce^(x)
Solution
This is a first-order linear equation. The integrating factor is e^(3x), leading to the solution y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
Learn More →
Showing 1 to 30 of 31 (2 Pages)
