Question

I want to double-check that my method is right so please post a detailed workup

Solve the IVP = 5 3 -3 :] [:) +(64*] []-[-] Y 1

Answer 1

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Solution Ginen I up is 63-6: 6315" (A -) za 5x+3y +6telt y'= -3x+ y · 210)=1,410)= | we use elimination method to solve it. differentiate eqn and substitute into it the derivative y from the ean ② -2t "=-5x+3y' +6e ~let ezt => 32"=-54'+36-32+y) +6ēt-iatēzt » «"=-54-9x+3y +6224_12t ezt we can solue the eqn 0 for 3y and substitute in the aboue ean 3y = x'tsa-ot ezt * *" = - 5'-9x + (x+51-6tē2b)+se2t 3) X"=-45-4x +62t-18tē =) ze"+4'+4x = 622t-18t ezt - t 3 This is second order linear homogeneous equation for the function act). puge Ly4

first solue the homogeneous equation- 20'+4x'+ 440 >> D + 4D +4) * =0 =) (0+2)? 20 The roots of the auseiliarly eq" are- a?+42+420 = (2+2)20 7 0,5-2, 12=-2 Then the solution of the homogeneous equation for alt) is as follows 4 Holt) = (stat)e²t where, c, and Ca are arbitrary constants. Now, we find out particular solution x,lt) dilt) = (6ēt-18722 (0+2) 7 2,1t) = 6 eet 18t ezt (0+2)2 +212 18 2t , t =) X,(t) = lottet (D-2+2)2 4,(t)= 3ť ezt -18 2t L(t) D2 = 2,(t) = 3+2 ezt -18 ezt 3 -) 2,1t)= 3+227 -18 22t t 6 2-25 e 3 m2 =72,1t) - 3t? 3t Page 2 of 4

6 Giuen that the general solution is given as- alt) = Holt) + x,1t) 74(t) = (0,+Ct)elt +372 24-34 ezt 210)=1, then put to in 6 210)= ((, ) e (i = 1, Hence H(t) = (ltet)ězt + 3té2t 3 ezt 7 It remains to find the function y(t), we calculate the derivative dellt) and substitute it into eqno of the original system d'lt)= -2(1+(at) e ²t + ca ezt + 6t e²t_6t² ezt 3 t e 9 t²e²t tot ezt =) -2 (1+(2+)ězt + szēzt + 60 ezt-67232t_9+2 ezt +6+32= -5[1146t)ělt + 3t 24-3622] +3y tot ezt -2t » -2 0 2t_z czte2t+cz e ²t + 6te + 5 ēzt +5c2tēzt +15ězt-15tet statt =34 => 34 = 3ēt +36ztēt + 5ēzt_943224 Ø Page 34

using initial condition y(0)=-1 3 410) = 3 + C = -3 = 3+cz = (=-6 Put C₂ =-6 in ① & @ final particular solution is- xelt) = (1-6t) e²+ + 34² e2t_at=2t 3y(t) = 3 e2t_18te²t_6e²t_ gq3 2t 2 39(t) = - 3e2t. -lotězt -g+302t => ylt) = - ezt-6tězt- 3+3 624 Page 4 of 4