Determine an annihilator for x2 e3x 5 cos 2xconvert the

Question

1. Solve the following differential equations:

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

2. Determine an annihilator for x² e^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^-2s /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.