Differential Equations
Q. Find the general solution of dy/dx = 3x^2. (2020)
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A.
y = x^3 + C
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B.
y = 3x^3 + C
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C.
y = x^2 + C
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D.
y = 3x + C
Solution
Integrating 3x^2 gives y = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. Find the general solution of the equation y' = 3x^2y.
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A.
y = Ce^(x^3)
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B.
y = Ce^(3x^3)
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C.
y = C/x^3
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D.
y = Cx^3
Solution
This is a separable equation. Integrating gives y = Ce^(x^3).
Correct Answer: A — y = Ce^(x^3)
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Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.
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A.
y = 2e^(4x)
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B.
y = e^(4x)
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C.
y = 4e^(x)
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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)
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Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.
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A.
y = 2e^(4x)
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B.
y = e^(4x)
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C.
y = 4e^(2x)
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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)
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Q. Find the solution of the differential equation dy/dx = y^2.
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A.
y = 1/(C - x)
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B.
y = C/(x - 1)
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C.
y = Cx
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D.
y = e^(x)
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. Find the solution of the differential equation y' = 3y + 6.
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A.
y = Ce^(3x) - 2
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B.
y = Ce^(3x) + 2
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C.
y = 2e^(3x)
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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
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Q. Find the solution of the equation dy/dx = y^2 - 1.
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A.
y = tan(x + C)
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B.
y = C/(1 - Cx)
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C.
y = 1/(C - x)
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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)
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Q. Find the solution of the equation y' + 2y = 0.
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A.
y = Ce^(-2x)
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B.
y = Ce^(2x)
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C.
y = 2Ce^x
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D.
y = Ce^x
Solution
This is a first-order linear differential equation. The solution is y = Ce^(-2x).
Correct Answer: A — y = Ce^(-2x)
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Q. Solve the differential equation dy/dx = 2x + 1.
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A.
y = x^2 + x + C
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B.
y = x^2 + 2x + C
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C.
y = 2x^2 + x + C
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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
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Q. Solve the differential equation dy/dx = 2y + 3. (2023)
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A.
y = Ce^(2x) - 3/2
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B.
y = Ce^(-2x) + 3/2
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C.
y = 3e^(2x)
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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
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Q. Solve the differential equation dy/dx = y/x. (2023)
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A.
y = Cx
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B.
y = Cx^2
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C.
y = C/x
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D.
y = C ln(x)
Solution
This is a separable equation. Integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. Solve the differential equation y' = 5 - 2y.
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A.
y = 5/2 + Ce^(-2x)
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B.
y = 5 + Ce^(-2x)
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C.
y = 2 + Ce^(2x)
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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)
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Q. Solve the differential equation y' = 5y + 3.
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A.
y = (3/5) + Ce^(5x)
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B.
y = (5/3) + Ce^(5x)
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C.
y = Ce^(5x) - 3
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D.
y = Ce^(3x) + 5
Solution
Using the integrating factor method, we find the solution y = (3/5) + Ce^(5x).
Correct Answer: A — y = (3/5) + Ce^(5x)
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Q. Solve the differential equation y'' - 3y' + 2y = 0.
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A.
y = C1e^(2x) + C2e^(x)
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B.
y = C1e^(x) + C2e^(2x)
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C.
y = C1e^(-x) + C2e^(-2x)
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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)
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Q. Solve the equation y' = 6y + 12.
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A.
y = 2 - Ce^(-6x)
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B.
y = Ce^(6x) - 2
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C.
y = 2 + Ce^(6x)
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D.
y = 6Ce^(-x)
Solution
This is a first-order linear equation. The integrating factor method gives the solution y = 2 - Ce^(-6x).
Correct Answer: A — y = 2 - Ce^(-6x)
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Q. Solve the first-order differential equation dy/dx = y/x.
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A.
y = Cx
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B.
y = Cx^2
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C.
y = C/x
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D.
y = C ln(x)
Solution
This is a separable equation. Separating variables and integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. Solve the first-order linear differential equation dy/dx + y/x = 1.
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A.
y = x + C/x
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B.
y = Cx - x
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C.
y = Cx + x
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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
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Q. What is the general solution of the equation y' = 4y + 3?
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A.
y = Ce^(4x) - 3/4
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B.
y = Ce^(4x) + 3/4
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C.
y = 3e^(4x)
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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
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Q. What is the general solution of the equation y'' - 3y' + 2y = 0?
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A.
y = C1 e^(x) + C2 e^(2x)
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B.
y = C1 e^(2x) + C2 e^(x)
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C.
y = C1 e^(3x) + C2 e^(0)
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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)
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Q. What is the general solution of the equation y'' - 4y' + 4y = 0?
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A.
y = (C1 + C2x)e^(2x)
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B.
y = C1 e^(2x) + C2 e^(-2x)
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C.
y = C1 e^(4x) + C2 e^(-4x)
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D.
y = C1 cos(2x) + C2 sin(2x)
Solution
The characteristic equation has a repeated root r = 2. The general solution is y = (C1 + C2x)e^(2x).
Correct Answer: A — y = (C1 + C2x)e^(2x)
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Q. What is the integrating factor for the equation dy/dx + (1/x)y = 2?
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A.
x
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B.
e^(ln(x))
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C.
e^(ln(x^2))
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D.
1/x
Solution
The integrating factor is e^(∫(1/x)dx) = e^(ln(x)) = x.
Correct Answer: C — e^(ln(x^2))
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Q. What is the integrating factor for the equation dy/dx + 2y = 3?
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A.
e^(2x)
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B.
e^(-2x)
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C.
e^(3x)
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D.
e^(-3x)
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
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Q. What is the integrating factor for the equation dy/dx + 2y = 6?
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A.
e^(2x)
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B.
e^(-2x)
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C.
e^(6x)
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D.
e^(-6x)
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
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Q. What is the particular solution of dy/dx = 4y with the initial condition y(0) = 2?
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A.
y = 2e^(4x)
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B.
y = e^(4x)
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C.
y = 4e^(4x)
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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)
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Q. What is the solution of the differential equation y' = 2y + 3?
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A.
y = Ce^(2x) - 3/2
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B.
y = Ce^(2x) + 3/2
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C.
y = 3e^(2x)
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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
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Q. What is the solution of the differential equation y' = 5y + 3?
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A.
y = (3/5) + Ce^(5x)
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B.
y = Ce^(5x) - (3/5)
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C.
y = (3/5)e^(5x)
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D.
y = Ce^(3x) + 5
Solution
Using the integrating factor method, we find the general solution to be y = Ce^(5x) - (3/5).
Correct Answer: B — y = Ce^(5x) - (3/5)
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Q. What is the solution of the equation dy/dx = 3x^2?
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A.
y = x^3 + C
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B.
y = 3x^3 + C
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C.
y = x^2 + C
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D.
y = 3x^2 + C
Solution
Integrating both sides gives y = ∫3x^2 dx = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. What is the solution of the equation dy/dx = 4y + 2? (2021)
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A.
y = Ce^(4x) - 1/2
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B.
y = Ce^(-4x) + 1/2
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C.
y = 2e^(4x) + C
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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
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Q. What is the solution of the equation dy/dx = 6 - 2y? (2021)
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A.
y = 3 - Ce^(-2x)
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B.
y = 3 + Ce^(-2x)
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C.
y = 2 - Ce^(2x)
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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)
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Q. What is the solution of the equation y' = -ky, where k is a constant?
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A.
y = Ce^(kt)
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B.
y = Ce^(-kt)
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C.
y = -Ce^(kt)
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D.
y = -Ce^(-kt)
Solution
This is a separable equation. Integrating gives y = Ce^(-kt).
Correct Answer: B — y = Ce^(-kt)
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