Solve Problem 4.82 (using Runge-Kutta method) for the forcing function

\[F(t)= \begin{cases}\frac{F_{0} t}{t_{1}} ; & 0 \leq t \leq t_{1} \\ F_{0}\left(\frac{t_{2}-t}{t_{2}-t_{1}}\right) ; & t_{1} \leq t \leq t_{2} \\ 0 ; & t \geq t_{2}\end{cases}\]

with \(F_{0}=2000 \mathrm{~N}, t_{1}=3 \mathrm{~s}\), and \(t_{2}=6 \mathrm{~s}\).

Data From Problem 4.82:-

Find the response of a damped single-degree-of-freedom system with the equation of motion \[m \ddot{x}+c \dot{x}+k x=F(t)\]
using Runge-Kutta method. Assume that \(m=5 \mathrm{~kg}, c=200 \mathrm{~N}-\mathrm{s} / \mathrm{m}, k=750 \mathrm{~N} / \mathrm{m}\), and \[F(t)= \begin{cases}\frac{F_{0} t}{t_{1}} ; & 0 \leq t \leq t_{1} \\ F_{0} ; & t \geq t_{1}\end{cases}\]
with \(F_{0}=2000 \mathrm{~N}\) and \(t_{1}=6 \mathrm{~s}\).