The dynamics of a forced spring-mass-damper system can be represented by the following second-order ODE:

md²x/dt² + cdx/dt + k₁x + k₃x³ = P cos(ωt)

where in m = 1 kg, c = 0.4 N · s/m, P = 0.5 N, and ω = 0.5/s. Use a numerical method to solve for displacement (x) and velocity (υ = dx/dt) as a function of time with the initial conditions x = υ = 0. Express your results graphically as time-series plots (x and υ verses t) and a phase plane plot (υ versus x). Perform simulations for both

(a) Linear (k₁ = 1; k₃ = 0) and

(b) Nonlinear (k₁ = 1; k₃ = 0.5) springs.