Task:

A. [Homogeneous Equations = Natural Oscillations] Suppose x = x(t). Solve the following homogeneous ordinary differential equations [mx"(t) +bx'(t) + kx(t) = 0] representing the 1- dimensional motion of a mass-spring system: 1. x"(t) + 2x'(t) + x(t) = 0; 2. x"(t) + 25x(t) = 0; 3. x"(t) + x'(t) + x(t) = 0; 4. X"(t) + 3x'(t) + 2x(t) = 0. B. [IVP = Initial Value Problems]-I Suppose x = x(t). Solve the following homogeneous IVPs. Take the upward direction as positive and the downward direction as negative. 1. x"(t) + 2x'(t) + x(t) = 0, x(0) = 1 mm, x'(0) = 0 mm/s; 2. x"(t) + 25x(t) = 0, x(0) = - 1 mm, x'(0) = 1 mm/s; 3. x"(t) + x'(t) + x(t) =0, x(0) = 1 mm, x'(0) = 0 mm/s; 4. x"(t) + 3x'(t) + 2x(t) = 0, x(0) = - 1 mm, x'(0) = 1 mm/s