Question

Consider the harmonic oscillator with friction given by (t)2(t)wr(t) = 0, kis the oscillation constant. We where I cR -> R and B > 0 is the friction constant and w0 m consider the case of weak damping given by w- p2>0. As you have checked in exercise 3 on sheet 1 the general solution to this equation is given by 2(t) — еxp(- Bt) [А. (wt)B sin (wt)] (3) COS where w this is a two-parameter family of solutions. /w-2 and A, BeR are constants. Since the constants A, B E R may be choosen freely, Please find the unique solution with initial data r(0) A, BE R in terms of xo, vo, w, wo and B. xo and (0) = vo by expressing the constants Hint: Use the formula for the first derivative (with the necessary modification of constants) from your solution of exercise 3 on sheet 1 Is the above solution periodic? In other words, does there exist a T0, such that r(tT) = x(t) for all t 0. Is there a T2 > 0, such that if the solution has a local maximum at to, then it has a local maximum at tktokT2 for all k e N? Hint: Sketch the above solution.

Answer 1

Aloulual)t Bsinl) e Px-PI A Cot tadt BAinla -B/ A Bi K A W -BAt BLO= -BMo Bu -१ A -Ah JA22 Cerfwt BA A whear Ag 1d A perio dic l Rilairr