When an object with a mass of $5$ kg is connected to the end of a spring, the spring extends $10$ meters and reaches equilibrium. This spring$-$mass system is placed in an environment with a damping constant of $5$ kilograms$/$second and the mass is released with no initial velocity by pulling it down $3$ meters from its equilibrium position. Find the function that shows the change in position of the given damped spring$-$mass system with time. Determine the period and amplitude of the movement. Determine how many seconds the system first reaches its maximum position $($in the positive or negative direction$)$ and the equilibrium position after t $=0.($Gravity $=10$ meters$/$second$)$ Then transform the equation of the mass$-$spring system. Convert it into a system of equations and draw a sample orbit by creating the vector field of the system and interpret the shape of this orbit. $($Draw the vector field and orbit$)$