Consider the equation y'' - y' - 2y = 0. (a) Show that y1(t) = e^-t and y2(t) = e^2t form a fundamental set of solutions. (b) Let y3(t) = -2e^2t , y4(t) = y1(t) + 2y2(t), and y5(t) = 2y1(t) - 2y3(t). Are y3(t), y4(t), and y5(t) also solutions of the given differential equation? (c) Determine whether each of the following pairs forms a fundamental set of solutions: [y1(t), y3(t)]; [y2(t), y3(t)]; [y1(t), y4(t)]; [y4(t), y5(t)].
A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of 2 lb.s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3 in/s, find its position u at any time t. Determine when the mass first returns to its equilibrium position.