A cart of mass 1 kg is joined to a wall by a spring of stiffness 4 N/m, and there is no friction…

Step 1. The particular part. The term $-2t\cos 2t$ is the particular integral. Its second derivative is $8\sin 2t + 8t\cos 2t$, so adding $4$ times itself leaves $8\sin 2t$.

Step 2. $A$. $x(0) = A = 0$.

Step 3. $B$. The velocity is $x' = 2B\cos 2t - 2\cos 2t + 4t\sin 2t$. At $t = 0$ this is $2B - 2 = 0$, so $B = 1$. Thus $x = \sin 2t - 2t\cos 2t$.

Step 4. Displacements. At $t = \dfrac{\pi}{2}$: $x = \sin\pi - \pi\cos\pi = \pi = 3.1416\ \text{m}$. At $t = \pi$: $x = \sin 2\pi - 2\pi\cos 2\pi = -2\pi = -6.2832\ \text{m}$.

Step 5. The velocity. With $B = 1$ we get $x' = 4t\sin 2t$. At $t = \dfrac{3\pi}{4}$: $x' = 3\pi\sin\dfrac{3\pi}{2} = -3\pi = -9.4248\ \text{m/s}$.