Consider the ODE u" + b(u)u' + g(u) = 0, where b and g are continuous functions, and b is positive. In system form, u' =

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Consider The Ode U B U U G U 0 Where B And G Are Continuous Functions And B Is Positive In System Form U 1 (26.38 KiB) Viewed 29 times

Consider the ODE u" + b(u)u' + g(u) = 0, where b and g are continuous functions, and b is positive. In system form, u' = = V v' = -b(u)v-g(u). (1) Let G(u) = f g(u) du be an antiderivative of g. Show that (a) the function V(u, v) = /v² +G(u) is a Lyapunov function for (1); (b) if G(u) → ∞o as u →∞, then all solutions are bounded for t > 0; (c) equilibrium points have the form (u*, 0) where u* is a root of g; (d) all possible w-limit sets are the equilibrium points.