Q. Find the general solution of the differential equation y'' - 5y' + 6y = 0.
-
A.
y = C1 e^(2x) + C2 e^(3x)
-
B.
y = C1 e^(3x) + C2 e^(2x)
-
C.
y = C1 e^(x) + C2 e^(2x)
-
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 cos(2x) = 0.
-
A.
x = (2n+1)π/4
-
B.
x = nπ/2
-
C.
x = (2n+1)π/2
-
D.
x = nπ
Solution
The general solution is x = (2n+1)π/4, where n is any integer.
Correct Answer:
A
— x = (2n+1)π/4
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Q. Find the general solution of the equation dy/dx = 3x^2y.
-
A.
y = Ce^(x^3)
-
B.
y = Ce^(3x^3)
-
C.
y = Ce^(x^3/3)
-
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. Find the general solution of the equation sin(x) + sin(2x) = 0.
-
A.
x = nπ
-
B.
x = nπ/2
-
C.
x = (2n+1)π/4
-
D.
x = nπ/3
Solution
Factoring gives sin(x)(1 + 2cos(x)) = 0, leading to x = nπ or cos(x) = -1/2.
Correct Answer:
A
— x = nπ
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Q. Find the general solution of the equation sin(x) + √3 cos(x) = 0.
-
A.
x = (2n+1)π/3
-
B.
x = (2n+1)π/6
-
C.
x = nπ
-
D.
x = (2n+1)π/4
Solution
The general solution is x = (2n+1)π/3, where n is an integer.
Correct Answer:
A
— x = (2n+1)π/3
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Q. Find the general solution of the equation sin(x) + √3cos(x) = 0.
-
A.
x = (2n+1)π/3
-
B.
x = nπ
-
C.
x = (2n+1)π/4
-
D.
x = nπ + π/6
Solution
The general solution is x = (2n+1)π/3, where n is an integer.
Correct Answer:
A
— x = (2n+1)π/3
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Q. Find the general solution of the equation sin(x) = -1/2.
-
A.
x = 7π/6 + 2nπ
-
B.
x = 11π/6 + 2nπ
-
C.
x = 7π/6, 11π/6
-
D.
Both 1 and 2
Solution
The general solutions are x = 7π/6 + 2nπ and x = 11π/6 + 2nπ.
Correct Answer:
D
— Both 1 and 2
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Q. Find the general solution of the equation sin(x) = sin(2x).
-
A.
x = nπ
-
B.
x = nπ/3
-
C.
x = nπ/2
-
D.
x = nπ/4
Solution
Using the identity sin(a) = sin(b) gives x = nπ or x = (2n+1)π/3.
Correct Answer:
A
— x = nπ
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Q. Find the general solution of the equation sin(x) = sin(π/4).
-
A.
x = nπ + (-1)^n π/4
-
B.
x = nπ + π/4
-
C.
x = nπ + 3π/4
-
D.
x = nπ + π/2
Solution
The general solution is x = nπ + (-1)^n π/4, where n is any integer.
Correct Answer:
A
— x = nπ + (-1)^n π/4
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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)
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Q. Find the general solution of the equation y' = 3y + 2.
-
A.
y = (C - 2/3)e^(3x)
-
B.
y = Ce^(3x) - 2/3
-
C.
y = 2/3 + Ce^(3x)
-
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 + 3.
-
A.
y = Ce^(5x) - 3/5
-
B.
y = Ce^(5x) + 3/5
-
C.
y = 3/5 + Ce^(-5x)
-
D.
y = 5x + C
Solution
The integrating factor method gives the general solution y = Ce^(5x) - 3/5.
Correct Answer:
A
— y = Ce^(5x) - 3/5
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Q. Find the general solution of the equation y'' - 3y' + 2y = 0.
-
A.
y = C1 e^(2x) + C2 e^(x)
-
B.
y = C1 e^(x) + C2 e^(2x)
-
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^(x) + C2 e^(2x).
Correct Answer:
B
— y = C1 e^(x) + C2 e^(2x)
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Q. Find the general solution of the equation y'' - 5y' + 6y = 0.
-
A.
y = C1 e^(2x) + C2 e^(3x)
-
B.
y = C1 e^(3x) + C2 e^(2x)
-
C.
y = C1 e^(x) + C2 e^(2x)
-
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 integral of (1/x) dx.
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
x + C
-
.
1/x + C
-
.
e^x + C
Solution
The integral of (1/x) is ln|x| + C, where C is the constant of integration.
Correct Answer:
A
— ln
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Q. Find the integral of (2x + 1)^3 dx. (2019)
-
A.
(1/4)(2x + 1)^4 + C
-
B.
(1/3)(2x + 1)^4 + C
-
C.
(1/5)(2x + 1)^4 + C
-
D.
(1/2)(2x + 1)^4 + C
Solution
Using substitution, the integral is (1/4)(2x + 1)^4 + C.
Correct Answer:
A
— (1/4)(2x + 1)^4 + C
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Q. Find the integral of (2x + 3)dx. (2022)
-
A.
x^2 + 3x + C
-
B.
x^2 + 3x + 1
-
C.
x^2 + 3 + C
-
D.
2x^2 + 3x + C
Solution
Integrating term by term: ∫2xdx = x^2 and ∫3dx = 3x. Thus, ∫(2x + 3)dx = x^2 + 3x + C.
Correct Answer:
A
— x^2 + 3x + C
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Q. Find the integral of cos(2x)dx. (2023)
-
A.
(1/2)sin(2x) + C
-
B.
sin(2x) + C
-
C.
(1/2)cos(2x) + C
-
D.
2sin(2x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx) + C. Here, k=2, so the integral is (1/2)sin(2x) + C.
Correct Answer:
A
— (1/2)sin(2x) + C
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Q. Find the integral of cos(x) with respect to x. (2023)
-
A.
sin(x) + C
-
B.
-sin(x) + C
-
C.
cos(x) + C
-
D.
-cos(x) + C
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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Q. Find the integral of cos(x). (2023)
-
A.
sin(x) + C
-
B.
-sin(x) + C
-
C.
cos(x) + C
-
D.
-cos(x) + C
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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Q. Find the integral of cos(x)dx. (2023)
-
A.
sin(x) + C
-
B.
-sin(x) + C
-
C.
cos(x) + C
-
D.
-cos(x) + C
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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Q. Find the integral of e^(2x) dx.
-
A.
(1/2)e^(2x) + C
-
B.
2e^(2x) + C
-
C.
e^(2x) + C
-
D.
(1/2)e^(x) + C
Solution
The integral of e^(2x) is (1/2)e^(2x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. Find the integral of e^x dx. (2022)
-
A.
e^x + C
-
B.
e^x
-
C.
x e^x + C
-
D.
ln(e^x) + C
Solution
The integral of e^x is e^x + C.
Correct Answer:
A
— e^x + C
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Q. Find the integral of f(x) = 2x + 3.
-
A.
x^2 + 3x + C
-
B.
x^2 + 3x
-
C.
x^2 + 3
-
D.
2x^2 + 3x + C
Solution
The integral ∫(2x + 3)dx = x^2 + 3x + C.
Correct Answer:
A
— x^2 + 3x + C
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Q. Find the integral of f(x) = 2x^3 - 4x + 1.
-
A.
(1/2)x^4 - 2x^2 + x + C
-
B.
(1/2)x^4 - 2x^2 + C
-
C.
(1/4)x^4 - 2x^2 + x + C
-
D.
(1/3)x^4 - 2x^2 + x + C
Solution
The integral ∫(2x^3 - 4x + 1)dx = (1/2)x^4 - 2x^2 + x + C.
Correct Answer:
A
— (1/2)x^4 - 2x^2 + x + C
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Q. Find the integral of sin(x) with respect to x. (2020)
-
A.
-cos(x) + C
-
B.
cos(x) + C
-
C.
sin(x) + C
-
D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer:
A
— -cos(x) + C
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Q. Find the integral of sin(x). (2020)
-
A.
-cos(x) + C
-
B.
cos(x) + C
-
C.
sin(x) + C
-
D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer:
A
— -cos(x) + C
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Q. Find the integral of sin(x)dx. (2020)
-
A.
-cos(x) + C
-
B.
cos(x) + C
-
C.
sin(x) + C
-
D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer:
A
— -cos(x) + C
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Q. Find the integral of x^2 with respect to x.
-
A.
(1/3)x^3 + C
-
B.
(1/2)x^3 + C
-
C.
(1/4)x^4 + C
-
D.
x^3 + C
Solution
The integral of x^2 is (1/3)x^3 + C, where C is the constant of integration.
Correct Answer:
A
— (1/3)x^3 + C
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Q. Find the integral of x^5 dx. (2020)
-
A.
(1/6)x^6 + C
-
B.
(1/5)x^6 + C
-
C.
(1/4)x^6 + C
-
D.
(1/7)x^6 + C
Solution
The integral is (1/6)x^6 + C.
Correct Answer:
B
— (1/5)x^6 + C
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