Determine an annihilator for x2e3x 5 cos

Question

1. Solve the following differential equations:

A. D (D +3)³ (D² +9)(D² -2x-8)y = 0
B. y^m-4y^m-9y'+36y=0
C. y^m-4y'= x +3 cos x + e^-2x [Hint: Use undetermined coefficients]
D. y" +4y' +4y = X^-2e^-2x with x > 0 [Hint: Use variation of parameters]

2. Determine an annihilator for x²e^3x + 5 cos 2x.

3. Convert the following system to a first-order system:

d²x/dt²-3dy/dt+x=sint,

d²y/dt² - tdx/dt-e'y = t2

4. Solve the following linear system of differential equations:

x1'= x1 + 2x₂
x₂'= 4x₁ + 3x₂

5. Solve the following nonhomogeneous linear system of differential equations:

x₁' = x₁+ 2x₂ +12e^3t
x₂' = 4x₁ +3x₂ +18e^2t

6. Compute the Laplace transforms of the following functions:

A. e^5t sin 3t
B. u₂ (t)sin (5 (t - 2))

7. Compute the inverse Laplace transforms of the following functions:

A. 7e^-21/s2 +4

B. 3s+2 /(s -1)(s -2)

8. Solve the following initial-value problem by using Laplace transforms:

yⁿ-4y=12e^2t with y(0) = 2 and y'(0) = 3.