Q. What is 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 + C)
Solution
Separating variables and integrating gives 1/y = x + C, leading to y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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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' + 4y = 0?
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A.
y = Ce^(-4x)
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B.
y = Ce^(4x)
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C.
y = 4Ce^x
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D.
y = Ce^(x/4)
Solution
This is a separable equation. The solution is y = Ce^(-4x).
Correct Answer: A — y = Ce^(-4x)
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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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Q. What is the solution to the differential equation dy/dx = -y/x?
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A.
y = Cx
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B.
y = C/x
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C.
y = Cx^2
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D.
y = Cx^(-1)
Solution
This is a separable equation. Separating variables and integrating gives y = C/x.
Correct Answer: B — y = C/x
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Q. What is the solution to 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 to be y = (3/5) + Ce^(5x).
Correct Answer: A — y = (3/5) + Ce^(5x)
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Q. What is the solution to the equation dy/dx = -5y?
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A.
y = Ce^(-5x)
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B.
y = -5Ce^x
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C.
y = Ce^(5x)
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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)
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Q. What is the solution to the equation dy/dx = y^2? (2022)
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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^2
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D.
y = ln(Cx)
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. What is the solution to 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 separable equation. The solution is y = Ce^(-2x).
Correct Answer: A — y = Ce^(-2x)
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Q. What is the solution to the equation y' + 3y = 0?
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A.
y = Ce^(-3x)
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B.
y = Ce^(3x)
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C.
y = 3Ce^(-x)
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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)
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Q. What is the solution to the 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 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
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Q. What is the solution to the equation y'' + 4y = 0?
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A.
y = C1 cos(2x) + C2 sin(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 sin(4x) + C2 cos(4x)
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The general solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. What is the solution to the equation y'' - 3y' + 2y = 0?
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A.
y = C1 e^(2x) + C2 e^(x)
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B.
y = C1 e^(x) + C2 e^(2x)
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C.
y = C1 e^(-x) + C2 e^(-2x)
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D.
y = C1 + C2x
Solution
The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2, leading to y = C1 e^(x) + C2 e^(2x).
Correct Answer: B — y = C1 e^(x) + C2 e^(2x)
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