Consider the differential equation dy/dt = f (y), where f (y) is a smooth function (it c...

Consider the differential equation dy/dt = f (y), where f (y) is a smooth function (it can be differentiated as many times as we like). Suppose y = y0 is an equilibrium point for this equation.

(a) Let u = y−y₀ and write the differential equation in terms of the new dependent variable u.

(b) Show that the linear approximation of the differential equation for du/dt near the equilibrium point at u = 0 is given by