For second order DES, the roots of the characteristic equation may be real or complex. If the roots are real, the complementary solution is the weighted sum of real exponentials. Use Cl and C2 for the weights, where Cl is associated with the root with smaller magnitude. If the roots are complex, the complementary solution is the weighted sum of complex conjugate exponentials, which can be written as a constant times a decaying exponential times a cosine with phase. Use Cl for the constant and Phi for the phase.

All numerical angles(phases) should be given in radian angles (not degrees). Given the differential equation
y'' + 7y' +12y = 7 cos(6t+ 0)u(t)

a. Find the functional form of the complementary solution, yc(t).
b. Find the particular solution, yp(t).
c. Find the total solution, y(t) for the initial condition y(0) = 1 and y'(0) = 12.