Question

question 5

Note. Some of the follow: notation x = dx/dt and following exercises use the overdot it and ï = d'x/dt- for first and second d also in Chapter 4, Section 1. ial equations 1 to 6 generates a free Cach (i.e., undriven) os classify each equation damped, underdamp e derivatives, used also in Chant of the differential equations 1 ven) oscillation. (a) Without solving it first, v each equation according to type: harmonic, over- underdamped, or critically damped. (b) Find the general solution formula for each equation. 1. d?x/dt2 + 2dx/dt + x = 0 2. dx/dt2 + 2dx/dt + 2x = 0 3. č + 9x = 0 4. č + 3i + x = 0 5. č + ax + a²x = 0, a > 0

Answer 1

has eq metbut ED we know that damped harmonic oscillator I mit but kn=o . with myo, bz, o and kyo A.E is mp²bD+k-0 ; D = -6 1624mk 2m is ir b<4mk 3 under damping 1) 6²7 4mk - overdamping 1) 6² 4mic critical damping. 1) Given D.E'S are s o + 2né + x 3050 vetene ten20 @ së tyre =0 @ * +32+x=0_ 0 nd ät are tare = 0 ao compair all D.E's ①, ③, 7 40 with ma-tbat kuzo, myo, byo, kto DIEO Here m=1>0,b=230, K=1>0 Mow, beze=4, 4mk = 4xIx/ hence nk= 4 = soitical damping

_ = 8 D. E Here mel>, b=2yo, le=2> Now,oo 1962= 2 =4, 4mk = 4X1X2 hence 6² 4mk under damping Die Here, mz_13063070, K-go --Now, 2 b²o. 4mk = 4X 1 X 9 S 36 hence b² <4mk = underdamping D.ED Here_mel, b=30, leclo How ? 325g Amk = 4x1*1 = 4 hence bez 4mk & overdamping DieņHere_mzl>0, brazo, k39230 a>0 low be=g², 4ml = 4x1x92 = 492 bence 62 <4ml = underdamping.

de 1. Given D. Eis 1 + 2 da tuzo " ütox tuzo. the Auxilary ean CAE) is D72D1-D (D+ D ost Jees . DE-L, -1 general solution of o is [ n(t) = (G+ at e

d2 2. dein +2 +2130 - * +236 +22=0 A 6 is 03728+2=0 :> D= -2 3 14-472 -=-12 éjt! general solution is act) = et (gost + esint 2 Given Dit_is_soprano ätgno A. E is 84930 ID = I3ė general solution is OCH = Goossttesinst 4. Given Pie is die +30+130 TAE is D²30+1=0 ------ D- 3LJ324x1 2

lanob tool- Good she | general scution is Fact) - 82 g cost tęsinto). (S. Given Die is xtant anzo gayo - i a A.Eis D²aDtq2_o 1 pc-9 Ja2492 =- 7951 general solution is | 16t9= elet( 9 cos se g t +e_sinugat) 21