Find the roots of the characteristic polynomial r and r-

Question

Consider the non-homogeneous differential equation

y" - 10 y' + 9 y = -3 cos(3t) + 5 sin(3t).

a. Find the roots of the characteristic polynomial, r₊ and r_-, larger and smaller or equal, respectively.

b. Find the real-valued fundamental solutions to the differential equation associated to the roots in part (a).

c. Use the functions in part (b) to find the general solution to the homogeneous equation, (use the names c and d for any arbitrary constants)

d. Find a particular solution y_p to the non-homogeneous equation.

e. Find the solution of the initial value problem y" - 10y' + 9y = -3 cos(3t) +5 sin(3t), y(0) = 2, y'(0) =-2.