Assignment:
Consider a harmonic oscillator with mass m = 1, spring constant k = 1, and damping coefficient b = 4. For the initial position y(O) = 2, find the initial velocity for which y(t) > 0 for all t and y(t) reaches 0.1 most quickly. [Hint: It helps to look at the phase plane first.]
Please consider the following second order differential equation.
d2y/dt2 + 4dy/dt + y=0
and the corresponding system dY/dt = CY, where C is a 2x2 matrix where c11 is 0, c12 is 1, c21 is -1, and c22 is -4. And Y is a 2x1 matrix where Y11 is y and Y21 is v.
Provide complete and step by step solution for the question and show calculations and use formulas.