4. A 2 kg mass is attached to a spring whose constant is 6 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 8 times the instantaneous velocity. (a) Write the second-order linear differential equation to model the motion. (b) Convert the second-order linear differential equation from part (a) to a first-order linear system. (e) Find the general solution. (d) Find the solution that satisfies the initial condition X (0) = ((0) ) = (²). ▪ Report your solution as one vector. (e) What type of motion does the solution of the initial value problem describe? (free undamped (simple harmonic) motion, free overdamped motion, free critically damped motion, or free underdamped (oscillatory) motion)

(1) A particle has an initial displacement of s = 30 metres when t = 0. The velocity of the particle is v= 6t² - 216t+ 1944 m/s (t > 0). (a) Find the displacement of the particle at time t. (b) Find the acceleration of the particle at time t. (c) What is the acceleration and displacement when v = 0?

Differential equations have been used extensively in the study of drug dissolution for patients given oral medications. The three simplest equations used are the zero-order kinetic equation, the Noyes-Whitney equation, and the Weibull equation. All assume that the initial concentration is zero but make different assumptions about how the concentration increases over time during the dissolution of the medication. The zero-order kinetic equation states that the rate of change in the concentration of drug c (in mg/mL) during dissolution is governed by the differential equation dc dt = k where k is a positive constant. Is this differential equation pure-time, autonomous, or nonautonomous? pure time autonomous nonautonomous State in words what this differential equation says about how drug dissolution occurs. The drug concentration increases linearly with time during dissolution. The drug concentration remains constant with time during dissolution. The drug concentration…