1. (a) The initial temperature of the finite rod 0 ≤ x ≤ is given by the function (x) = sin(2x). The ends of the rod are kept at temperature zero for all times, and the rod is heated permanently, with the heat source taking the form f(t) = et. Find the temperature distribution in the rod for t> 0 by solving the problem ut = a²uzz + e-t, u(t,0) = u(t, π) = 0, u(0, x) = sin(2x), 0 0, 0≤x≤ π.

= A mass m = 4 kg is attached to both a spring with spring constant k 577 N/m and a dash-pot with damping constant c = 4 N. s/m. The mass is started in motion with initial position o 3 m and initial velocity = 5 m/s. Determine the position function z(t) in meters. x(t) Note that, in this problem, the motion of the spring is underdamped, therefore the solution can be written in the form r(t) Ciecos (wit-a₁). Determine C₁, w₁,ajand p. C₁ WI XX1 P = 11 Co (e^(-1/2t))[((-5/6)sin(12t))+(3cos(12t))] Graph the function z(t) together with the "amplitude envelope" curves z-C₁e- and = ₁e pt. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so c= 0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t)= Cocos (wotao). Determine Co. we and co. = ولما (assume 0 ≤ a <2m)

Find the maximum and minimum values of the function g(0) = 60 – 9 sin(0) on the interval - [0. =] 2 Minimum value Maximum value =