Avery simple mechanical oscillator can be modeled using the second-order differential equation d2x dt2 + dx

Avery simple mechanical oscillator can be modeled using the second-order differential equation d2x dt2 + μ

dx dt + 25x = 0, where x measures displacement from equilibrium.

(a) Rewrite this equation as a linear first-order system by setting ˙ x = y.

(b) Sketch phase portraits when (i) μ = −8, (ii) μ = 0, (iii) μ = 8, and

(iv) μ = 26.

(c) Describe the dynamical behavior in each case given that x(0) = 1 and

˙ x(0) = 0.

Plot the corresponding solutions in the tx plane.