Question

Solve the given initial value problem. x(0) = 1 dx = 4x +y- e 3t, dt dy = 2x + 3y; dt y(0) = -3 The solution is X(t) = and y(t) =

Answer 1

Given da de & 40ty 8t 2000 l dy - 2x +39 yo) =-3 U Now Desire differential equation in a tema. da = 40 ty at differenciate with wespect to 4 da dt t dy P 3e 3t from 92 dy -24 = 2.7 1 a €93 - 3 X 600 do des Bt f dy -ze 3 da 3t -zy +ze d -zda dt = 4 da dt² + dy -B4 d2a 3da dt a da dt2 +12 - 2x + 2 9 da dt + 10 20 d)

Given = 4x + y est - 2x toy for understanding early. destining differential ean in "y" tema Now Apply differenciate with respect to t both En au 42'+y! 8+ yu = 2x + 34' 89 2 Again apply differentation respect to "t" 2 Suosteeti Bq ② and 690 -34 3e8t -zyn - 44" + +12y! gulay +124 +y

Hence both the differential cana have bean derived. first fend general solution differential 89nin a-temmx. 24 9x tox-O — 2D +10 find the roots. -50-20 + 10 20 6 (0-5)-2CD-s) to D=21 and bas the wots ano 2 and 5 roots are real and un equal for real and un equal oot the general son in 4 e Bot at at the Bt a G toest се . test y tt) 22+ tore st Rect) = catto est]

General solution to differential son in y-terma yuldu +124" tese me have to find Ip and Gh. General solution Intup! D3 - 79² 4 2020 DCD? - AD + 12 ) zo - 46-30 112 ro D (0-d)-3CD-1) = (o-A) (0-3) 20 De a) and (o=31 and 0:01 The roots Becal un equal. 4t ce & Get + Cz é 4t a test teze yp = А е PE BA est 9e3t Gue HT oot 2AAE 29: ditterential au Sp's Sub

3+ 27 AC (GABt) + 12 (3 A03t -6e3t - 63A -1 361) 3t -6e3t (27 A (63A - 63A) 3+ 6036 How multiply Ip with = Astlet Yp = Acest +3x 3 x 34 ) esp= A ( 3 3 4 383 490 e 3t ) cast are 3+ 3+ -1 + 27 x et sub in DE 3t + 24x est ) - ( 6e3+ +98 et At 2 ( 3 4 3 x 3 ) A = -68 3t 2DA 3+ + 27 A se 3 420 A - 63a e 3 + + 4 +367 3t A =-60 3t 394 est -4 2003t +63AX 3 - 63 A Rest 3t -Bne A2 2x est General souction y z Intep st a toe tege at 37 + 20e

Given yo) = -3 and acol=1 Geet & Gest y ct) = at Gest 4. 3t + 2x and Get ca eo -3= a theo theo ata = a test z = -3 1 + = -3 C3 = - 4 we cant find a and a 5t と -C₂ and yeto 4t st taxe - 4 e