Given differential equation is

\(\frac{d^2y}{dt}^2 + 6 \frac{dy}{dt} + 8y = 5 sin\,3t\)

\(\Rightarrow\) D²y + 6Dy + 8y = 5 sin 3t (\(\because \frac{d}{dt} = D\) & \(\frac{d^2}{dt^2} = D^2\))

\(\Rightarrow\) (D² + 6D + 8)y = 5sin 3t

The characterstic equation of given differential equation is

\(\Rightarrow\) m² + 6m + 8

\(\Rightarrow\) m² + 2m + 4m + 8 = 0

\(\Rightarrow\) m(m + 2) + 4(m + 4) = 0

\(\Rightarrow\) (m + 2)(m + 4) = 0

\(\Rightarrow\) m = -2, -4

The complementary function is

C. F = C₁e^-2t + C₂e^-4t

Now, particular integral is

P.I = \(\frac{1}{D^2 + 6D + 8} \) 5 sin 3t

= \(\frac{1}{-9 + 6D + 8}\) 5 sin 3t (\(\frac{1}{f(D)^2} = \frac{1}{f(-a)^2} sin ax; \, f(-a^2) \neq 0\))

= \(\frac{6D + 1}{(6D - 1)(6D + 1)}\) 5sin 3t

= \(\frac{6D + 1}{36D^2 - 1}\) 5 sin 3t

= 5 (6D + 1) \(\frac{sin \,3t}{-36 \times 9-1}\)

= \(\frac{-5}{325}\) (6D + 1) sin 3t

= \(-\frac{1}{65} (18 cos\, 3t + sin \, 3t)\)

Hence the solution is y = C.F + P.I

 \(\Rightarrow\) y = C₁e^-2t + C₂e^-4t - \(\frac{1}{65}\)(18 cos 3t + sin 3t)