MATH 3321 Quiz 8 10/2/14

1. A particular solution of y′′ + 9y′ = 4cos 3x + 3 sin 2x + 2 will have the form:

(a) z = A cos 3x + B sin 3x + C cos 2x + D sin 2x + Ex + F

(b) z = Ax cos 3x + Bx sin 3x + C cos 2x + D sin 2x + Ex

(c) z = A cos 3x + B sin 3x + C cos 2x + D sin 2x + Ex

(d) z = A cos 3x + B sin 3x + C cos 2x + D sin 2x + E

(e) None of the above.

2. A particular solution of y′′ + 4y′ + 4y = 4e−2x + 2xe2x will have the form:

(a) z = (Ax3 + Bx2 + Cx)e−2x + (Dx + E)e2x

(b) z = Ax3e−2x + Bxe2x

(c) z = (Ax2 + Bx)e−2x + (Cx + D)e2x

(d) z = Ax2e−2x + (Bx + C)e2x

(e) None of the above.

3. A particular solution of y′′ + 4y′ + 4y = 4xe−2x + 2e2x will have the form:

(a) z = (Ax3 + Bx2 + Cx)e−2x + (Dx + E)e2x

(b) z = (Ax3 + Bx2)e−2x + Ce2x

(c) z = (Ax2 + Bx)e−2x + Cxe2x

(d) z = Ax3e−2x + Be2x

(e) None of the above.

4. A particular solution of y′′− 2y′

− 8y = 2e−x cos 3x− (3x2 + 2x)e−2x + 4e4x will have theform

(a) z = Ae−x cos 3x + Be−x sin 3x + (Cx2 + Dx + E)e−2x + Fxe4x

(b) z = Ae−x cos 3x + Be−x sin 3x + (Cx3 + Dx2 + Ex)e−2x + Fe4x

(c) z = Ae−x cos 3x + Be−x sin 3x + (Cx3 + Dx2 + Ex)e−2x + Fxe4x

(d) z = Ae−x cos 3x + Be−x sin 3x + (Cx3 + Dx2)e−2x + (Ex + F )e4x

(e) None of the above.

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5. The general solution of y′′− 4y′ + 4y = 2e2x +

e2x

x2is:

(a) y = C1e2x + C2xe2x + e2x ln x + x2e2x

(b) y = C1e2x + C2e

−2x− xe2x ln x − xe2x

(c) y = C1e2x + C2xe2x + xe2x ln x + x2e2x

(d) y = C1e2x + C2xe2x

−

e2x ln x

x2− x2e2x

(e) None of the above.

6. The general solution of y′′ + 4y = 4e2x + 4 sec2 2x is:

(a) z = C1 cos 2x + C2 sin 2x + 1

2xe2x + 1

2sin 2x ln(sec 2x + tan 2x)

(b) z = C1 cos 2x + C2 sin 2x + 1

2e2x + sin 2x ln(sec 2x + tan 2x) − 1

(c) z = C1 cos 2x + C2 sin 2x + e2x + cos 2x ln(sec 2x + tan 2x) − 1

(d) z = C1 cos 2x + C2 sin 2x + e2x + 1

2cos 2x ln(sec 2x + tan 2x)

(e) None of the above.

7. An object in simple harmonic motion has period 1

4π. At time t = 0, y(0) = 2, y′(0) = 0.

The equation of motion is:

(a) y = 2 sin(

4t + 1

2π

)

(b) y = 4 sin(

8t + 1

4π

)

(c) y = 2 sin(

8t + 1

2π

)

(d) y = sin(

4t + 1

4π

)

(e) None of the above.

8. An object in simple harmonic motion has period 1

2π. At time t = 0, y(0) = 0, y′(0) = 2.

The amplitude is:

(a) A = 2

(b) A = 4

(c) A = 1

2

(d) A = 1

4

(e) None of the above.

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9. y′′ + 4y′ + 20y = 3cos 4x is the mathematical model for a mechanical vibrating system.The transient solution is:

(a) y = C1e−4x cos 2x + C2e

−4x sin 2x

(b) y = C1e−2x cos 4x + C2e

−2x sin 4x + 1

2cos 4x + 2 sin 4x

(c) y = cos 4x + 4 sin 4x

(d) y = C1e−2x cos 4x + C2e

−2x sin 4x

(e) None of the above.

10. y′′ + 2y′ + 10y = 6 sin 2x is the mathematical model for a mechanical vibrating system.The steady state solution is:

(a) z = C1e−x cos 3x + C2e

−x sin 3x

(b) z = C1e−x cos 3x + C2e

−x sin 3x − 12 cos 2x + 18 sin 2x

(c) z = 18 sin 2x − 12 cos 2x

(d) z = 3 sin 2x + 2cos 2x

(e) None of the above.

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