Question

Problem 3.18 (Resonance catastrophe). Solve the equation xcos(wt), wo,w> 0. Discuss the behavior of solutions as t oo. The inhomogeneous term is also known as a forcing term. It is resonant ifw wo. What happens in this case?

Answer 1

Given that it wonacos (wt) , wooo It is a inhomogenious secon order lineal differential equation quation with constant Gefficients, Hence the general solution in X(t) = xetxo consider ä tworzo (Ditwó)x=0 → fen). 2 = 0 The auxiliary equation is tem) ao the = masiwa roots are complex conjugates. xc =qcos (Wot) tai sin (wot) Coswt op = P cos feos oet word Replaed by a ۱ ما بر -wat wa wow to : alth = Xc of up 1 x (H = a, cos (wot) & G sin (Wot) + coset / which is the general solution of an O. As - 0 xlt) D. Hence the behavious of the solutions approches to infinity as t o.

It w a wo, we get xp = cos (wat). Dit was : 1 2 Replace ă by - wor, we get Cos(wot) It is not possible) twe 2D apat cos (Wot) t. I. Cos (Wot) Here Daha - t. ( cas (wot) at xp = t. sin (wot) i xlt - C, Cos (wo t) + C sin wot + tsinloot) It wawo then the solation dren is 2CH = G, cafest) + C sin(at) + ta sin cost) Please give positive ratings. Thank you.