Differential Equations
Q. Find the general solution of 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. Integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer: A — y = Ce^(2x)
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Q. Find the general solution of the differential equation dy/dx = y.
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A.
y = Ce^x
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B.
y = Ce^(-x)
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C.
y = Cx
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D.
y = C/x
Solution
This is a separable equation. Integrating gives ln|y| = x + C, hence y = Ce^x.
Correct Answer: A — y = Ce^x
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Q. Find the general solution of the differential equation y'' - 5y' + 6y = 0.
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A.
y = C1 e^(2x) + C2 e^(3x)
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B.
y = C1 e^(3x) + C2 e^(2x)
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C.
y = C1 e^(x) + C2 e^(2x)
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D.
y = C1 e^(4x) + C2 e^(5x)
Solution
The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).
Correct Answer: B — y = C1 e^(3x) + C2 e^(2x)
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Q. Find the general solution of the equation y' = 3y + 2.
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A.
y = (C - 2/3)e^(3x)
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B.
y = Ce^(3x) - 2/3
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C.
y = 2/3 + Ce^(3x)
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D.
y = 3x + C
Solution
This is a first-order linear differential equation. The integrating factor is e^(-3x).
Correct Answer: B — y = Ce^(3x) - 2/3
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Q. Find the general solution of the equation y'' - 5y' + 6y = 0.
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A.
y = C1 e^(2x) + C2 e^(3x)
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B.
y = C1 e^(3x) + C2 e^(2x)
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C.
y = C1 e^(x) + C2 e^(2x)
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D.
y = C1 e^(4x) + C2 e^(5x)
Solution
The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).
Correct Answer: B — y = C1 e^(3x) + C2 e^(2x)
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Q. Find the particular solution of dy/dx = 2x with the initial condition y(0) = 1.
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A.
y = x^2 + 1
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B.
y = x^2 - 1
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C.
y = 2x + 1
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D.
y = 2x - 1
Solution
Integrating gives y = x^2 + C. Using the initial condition y(0) = 1, we find C = 1.
Correct Answer: A — y = x^2 + 1
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Q. Find the particular solution of dy/dx = x + y, given y(0) = 1.
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A.
y = e^x + 1
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B.
y = e^x - 1
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C.
y = x + 1
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D.
y = x + e^x
Solution
The general solution is y = e^x + C. Using the initial condition y(0) = 1, we find C = 1.
Correct Answer: A — y = e^x + 1
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Q. Find 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 = 3/2 - Ce^(2x)
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D.
y = 3/2 + Ce^(-2x)
Solution
This is a linear first-order equation. The general solution is y = 3/2 + Ce^(-2x).
Correct Answer: D — y = 3/2 + Ce^(-2x)
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Q. Find the solution of the differential 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^(x) + C2 e^(-x)
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D.
y = C1 sin(2x) + C2 cos(2x)
Solution
This is a second-order linear homogeneous differential equation. The characteristic equation has roots ±2i.
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. Find the solution of the first-order linear differential equation dy/dx + y = e^x.
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A.
y = e^x + Ce^(-x)
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B.
y = e^x - Ce^(-x)
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C.
y = e^(-x) + Ce^x
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D.
y = e^(-x) - Ce^x
Solution
Using an integrating factor e^x, we solve to get y = e^x + Ce^(-x).
Correct Answer: A — y = e^x + Ce^(-x)
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Q. Solve the differential equation dy/dx + 2y = 4.
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A.
y = 2 - Ce^(-2x)
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B.
y = 2 + Ce^(-2x)
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C.
y = 4 - Ce^(-2x)
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D.
y = 4 + Ce^(2x)
Solution
This is a linear first-order differential equation. The integrating factor is e^(2x). Solving gives y = 2 - Ce^(-2x).
Correct Answer: A — y = 2 - Ce^(-2x)
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Q. Solve 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 + C
Solution
Integrating both sides gives y = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. Solve the differential equation dy/dx = x^2 + y^2.
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A.
y = x^3/3 + C
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B.
y = x^2 + C
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C.
y = x^2 + x + C
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D.
y = Cx^2 + C
Solution
This is a non-linear differential equation. The solution can be found using substitution methods.
Correct Answer: A — y = x^3/3 + C
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Q. Solve 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 = 3e^(3x) + 2
Solution
Using the integrating factor method, we find y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
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Q. Solve the differential 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 cos(x) + C2 sin(x)
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D.
y = C1 e^(x) + C2 e^(-x)
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
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Q. Solve the differential equation y'' - 5y' + 6y = 0.
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A.
y = C1 e^(2x) + C2 e^(3x)
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B.
y = C1 e^(3x) + C2 e^(2x)
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C.
y = C1 e^(x) + C2 e^(2x)
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D.
y = C1 e^(2x) + C2 e^(x)
Solution
The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).
Correct Answer: B — y = C1 e^(3x) + C2 e^(2x)
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Q. Solve the equation dy/dx = y^2 - x.
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A.
y = sqrt(x + C)
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B.
y = x + C
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C.
y = 1/(C - x)
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D.
y = x - C
Solution
This is a separable equation. Separating variables and integrating gives y = 1/(C - x).
Correct Answer: C — y = 1/(C - x)
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Q. Solve the equation y' = y(1 - y).
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A.
y = 1/(C - x)
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B.
y = 1/(C + x)
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C.
y = C/(1 + x)
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D.
y = C/(1 - x)
Solution
Separating variables and integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. Solve the first-order linear differential equation dy/dx + y/x = x.
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A.
y = x^2 + C/x
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B.
y = Cx^2 + x
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C.
y = C/x + x^2
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D.
y = x^2 + C
Solution
Using the integrating factor e^(∫(1/x)dx) = x, we can solve the equation.
Correct Answer: A — y = x^2 + C/x
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Q. What is the general solution of the differential equation dy/dx = 3y?
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A.
y = Ce^(3x)
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B.
y = Ce^(-3x)
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C.
y = 3x + C
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D.
y = Cx^3
Solution
The differential equation is separable. Integrating both sides gives ln|y| = 3x + C, hence y = Ce^(3x).
Correct Answer: A — y = Ce^(3x)
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Q. What is the integrating factor for the equation dy/dx + 2y = 3x?
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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 + 3y = 6x?
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A.
e^(3x)
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B.
e^(-3x)
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C.
e^(6x)
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D.
e^(-6x)
Solution
The integrating factor is e^(∫3dx) = e^(3x).
Correct Answer: A — e^(3x)
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Q. What is the particular solution of dy/dx = 4x with the initial condition y(0) = 1?
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A.
y = 2x^2 + 1
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B.
y = 4x^2 + 1
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C.
y = 2x^2
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D.
y = 4x^2 + C
Solution
Integrating gives y = 2x^2 + C. Using the initial condition, C = 1.
Correct Answer: A — y = 2x^2 + 1
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Q. What is the particular solution of the equation dy/dx = 2y with the initial condition y(0) = 1?
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A.
y = e^(2x)
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B.
y = e^(2x) + 1
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C.
y = 1 + e^(2x)
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D.
y = 1 + 2x
Solution
The general solution is y = Ce^(2x). Using the initial condition y(0) = 1, we find C = 1.
Correct Answer: A — y = e^(2x)
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Q. What is the particular solution of the equation dy/dx = 2y with y(0) = 5?
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A.
y = 5e^(2x)
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B.
y = 2e^(2x)
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C.
y = 5e^(-2x)
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D.
y = 5 + 2x
Solution
The general solution is y = Ce^(2x). Using y(0) = 5 gives C = 5.
Correct Answer: A — y = 5e^(2x)
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Q. What is the solution of the differential equation dy/dx = (x^2 + 1)y?
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A.
y = Ce^(x^3/3 + x)
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B.
y = Ce^(x^2 + 1)
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C.
y = Ce^(x^2/2)
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D.
y = Ce^(x^3)
Solution
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
Correct Answer: A — y = Ce^(x^3/3 + x)
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Q. What is the 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, leading to the solution form (C1 + C2x)e^(2x).
Correct Answer: A — y = (C1 + C2x)e^(2x)
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Q. What is the solution to the differential equation dy/dx = (x^2 + 1)y?
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A.
y = Ce^(x^3/3 + x)
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B.
y = Ce^(x^2 + 1)
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C.
y = Ce^(x^2/2)
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D.
y = Ce^(x^3)
Solution
This is a separable equation. Integrating gives y = Ce^(x^3/3 + x).
Correct Answer: A — y = Ce^(x^3/3 + x)
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Q. What is the solution to the differential equation dy/dx = 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 = 3/2e^(2x)
Solution
Using the integrating factor method, we find y = Ce^(2x) + 3/2.
Correct Answer: B — y = Ce^(2x) + 3/2
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Q. What is the solution to 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 y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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