Solve the following initial value problem.
\[
\frac{d^{2} s}{d t^{2}}=-9 \cos (3 t+\pi), s^{\prime}(0)=400, s(0)=0
\]
\[
\mathrm{s}=
\]
(Type an exact answer, using $\pi$ as needed.)

Step 1 ：The given differential equation is a second order linear homogeneous differential equation with constant coefficients. The general solution of such an equation is given by the integral of the right hand side function.

Step 2 ：First, we integrate the right hand side function \(-9 \cos (3 t+\pi)\) with respect to \(t\).

Step 3 ：\[ \int -9 \cos (3 t+\pi) dt = -3 \sin (3 t + \pi) + C_1 \]

Step 4 ：Then, we integrate the result again to get the general solution of the differential equation.

Step 5 ：\[ \int -3 \sin (3 t + \pi) dt = \cos (3 t + \pi) + C_2 \]

Step 6 ：Now, we apply the initial conditions \(s^{\prime}(0)=400\) and \(s(0)=0\) to find the constants \(C_1\) and \(C_2\).

Step 7 ：Substituting \(t=0\) into the first integral, we get \(C_1 = 400\).

Step 8 ：Substituting \(t=0\) into the second integral, we get \(C_2 = 0\).

Step 9 ：Therefore, the solution of the initial value problem is

Step 10 ：\[ s(t) = \cos (3 t + \pi) + 400t \]

Step 11 ：Finally, we check whether this solution satisfies the initial conditions. Substituting \(t=0\) into the solution, we get \(s(0) = 0\), which is consistent with the initial condition. Differentiating the solution and substituting \(t=0\), we get \(s^{\prime}(0) = 400\), which is also consistent with the initial condition.

Step 12 ：Therefore, the solution of the initial value problem is correct.