Sol.4.3 - 4.3 Exercises we the initial conditions, soivethe...

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Exercises 4.3 -,tisfy the initial conditions, we soive the system 2 :31(o):c, -17 12 : a'(0) : - (c112) - (5c212) - *ftrL'e, t lte attsr.t'et' is t'2 : J. a : e,-tt212 cosh(st/2) - 3 sinh(5t/2)l . if S f .3: Auxiliary Equations with Complex Roots .-,,::-if\.eqlation in this problem is r'2 + 1 :0, which has roots r: td' \&'e see - ,i Altd 0 : I. Thus, a gerreral solntion to the clifferential eqrtation is given bv Uft): c1g(0)rcosf * c2e\lo)t sinf : cl cost * c2 sint. ,rr' eeuation, 12 - 10r .126:0, lias roots r': 5 * i' So, 'Y : 5, /J: 1' arid AQ) : c1e5t cos t * czebt sirtt -1 solittiou. r-errtial equation has the auxiliary equation 12 - 4r'* 7 : 0. The roots of -ar,- equation are ?. : (4 + fi6 -m) lZ : 2 + JSi. \\re see that o : 2 and Thus, a general solution to the differential eqttation is given by . ,t / ,- ' / - \ u'(l) : c1e2lcos (/lt; + "re'' sin (r/3t,) . -2 + J4 -r4 roots, r: -2!2i. by 4 solution is given / r.\ r/2si'{g} \2/ iras trn'o coniplex -:ar'1' ecluation for this problem is given by =,'*6:0 + 2r2+2rf3:0 + :. n : -ll2antl p : t/5l2,attd a generarl / r.\ aU) : cp-tt2,"' ( # ) *,,," \" / r iated aLrxiliary ecluation, r'2 + 4'r * 8 : 0, r iulswet iS -f; I VJ . -
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Chapter 4 12. The auxiliary equation for this problem is 12 + 7:0 rvith roots r: ttfri. Hence, u(/): {'1 cos (ttt,l112sitt (/7,\ '\/-\/ where c1 and c2 dre arbitrary constatrts, is a general solutiorr.
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Sol.4.3 - 4.3 Exercises we the initial conditions, soivethe...

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