Q. Determine the solution of the differential equation dy/dx = y^2 - 1.
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A.
y = tan(x + C)
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B.
y = 1/(C - x)
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C.
y = 1/(C + x)
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D.
y = e^(x + C)
Solution
This is separable. Separating and integrating gives y = 1/(C - x).
Correct Answer: B — y = 1/(C - x)
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Q. Find the general solution of the equation dy/dx = 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 = Ce^(x^3/3)
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D.
y = Ce^(x^2)
Solution
This is a separable equation. Separating and integrating gives y = Ce^(x^3).
Correct Answer: A — y = Ce^(x^3)
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Q. Solve the differential equation dy/dx = 2y.
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A.
y = Ce^(2x)
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B.
y = 2Ce^x
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C.
y = Ce^(x/2)
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D.
y = 2x + C
Solution
This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer: A — y = Ce^(2x)
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Q. Solve the differential equation dy/dx = 5 - 2y.
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A.
y = 5/2 + Ce^(-2x)
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B.
y = 5/2 - Ce^(-2x)
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C.
y = 2.5 + Ce^(2x)
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D.
y = 2.5 - Ce^(2x)
Solution
Rearranging gives dy/(5 - 2y) = dx. Integrating both sides leads to y = 5/2 + Ce^(-2x).
Correct Answer: A — y = 5/2 + Ce^(-2x)
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Q. What is the general solution of the differential 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 integrating factor for the equation dy/dx + 3y = 6?
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A.
e^(3x)
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B.
e^(-3x)
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C.
3e^(3x)
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D.
3e^(-3x)
Solution
The integrating factor is e^(∫3dx) = e^(3x).
Correct Answer: A — e^(3x)
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Q. What is the solution of the 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 separable. Integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. What is the solution to the differential equation dy/dx = xy?
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A.
y = Ce^(x^2/2)
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B.
y = Ce^(-x^2/2)
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C.
y = Cx^2
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D.
y = C/x
Solution
This is separable. Separating and integrating gives y = Ce^(x^2/2).
Correct Answer: A — y = Ce^(x^2/2)
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Q. What is the solution to the initial value problem dy/dx = 4y, y(1) = 2?
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A.
y = 2e^(4x)
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B.
y = 2e^(4x-4)
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C.
y = e^(4x)
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D.
y = 4e^(x)
Solution
The general solution is y = Ce^(4x). Using y(1) = 2, we find C = 2e^(-4).
Correct Answer: B — y = 2e^(4x-4)
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