S(t), t >= 0 be a geometric Brownian motion process with drift parameter mu=0.1 and volatility parameter σ=0.2.

Activity 3: The car body vertical displacement, y(t), can be approximately modelled as a damped mass-spring system (called car suspension system) using the following differential equation: Car Body Attaches Here Spring and Damper Hub Assembly ННИНЯТ d²y + b₂ dt² Wheel and Tire dy dt + ky = f(t) k Car body mcar 4 bv y (t) f(t) 7/1 Where y(t) is the vertical displacement, mear is the car mas, b, is the damping coefficient, k is the spring stiffness constant, t is the time, and f(t) is the hight of the street from the ground level as a function of time. Given that m = 1500 (Kg), b₂ = 2000 (N.s/m), and k = 100000 (N/m). Answer the following questions: (3-A) If the street is straight, so the f(t) = 0, determine the general solution of y(t). (3-B) Given the initial conditions y(0) = 2, and y'(0) = 1, determine the particular solution of y(t). (3-C) Use Laplace transform to solve the given differential equation with the same initial conditions given in (3-B).

The motion of a mass-spring system with damping is governed by y''(t) + by' (t) +9y(t) = 0; y(0) = 3, and y'(0) = 0. Find the equation of motion and sketch its graph for b = 0, 2, 6, and 10. What is the equation of motion for b = 0? y(t) = 3 cos 3t (Type an exact answer, using radicals as needed.) What is the equation of motion for b = 2? y(t) = (Type an exact answer, using radicals as needed.) ( ・・・ )

(18) Consider a particle with position function 7 (t) = (t, t² – 3t, 2t2). (a) Find the velocity, acceleration and speed of the particle. (b) What can you say about maximum and minimum values of the speed of this particle?