Question

Solve the given initial value problem by undetermined coefficients (annihilator approach). Prime not power for (3)

y^(3) + 9y' = e^x cos(3x)

y(0) = 2

y' (0) = 1

y''(0) = 1

Answer 1

value problem is sol" :- . Given initial + g.y! = en ces (3x) yo) 22, y! (0)=1, y"(0) = 1, can be written y (3) with equation o m 2 (6 = (63 +90) y és (34) Now the auxiliary equation is 2 3 m?+ 9m m=0 m#9 ire. m=o, I3i m= È Bi the ane 2 Hence complementary function is cof= i + G (os.3x + C₂ sin 3x G ,, cons tante corresporting to special form (²) ces (3x) of Ritis of 6 choose trical solution for Pit. ex (Asin 3x + 13 055 3x). Then the particular solution is taken on (Asim 3s + B Cos 3u) we ara * as en -3 k e Col3x) Since ya satisfies we hane D3 +90) y* en cas on (D3 ye+ + 9 Dyu) sir3nt B S3n) tee (3 A (ann 3B sinismo 2 Now x By*= e ( A

[vt4}s (29- 5-) + +8 ) (+ - + 9)] : é 8+ x 29 (1) - +5) + 4-15 (9-17)-13 - 24 [vsus 49 -) + {5) (4s - 99) [((36-+) +ws - svc+8):-) theo Rais = 2* (6f24) x 3 1 + (- 3g) ++ + + * LP th) 6 31 + (A- 30) s s , e 3Y

p3 you ee [6264 +18B) sin 3x +(184 - 268) Cos 34 Now putting there e* [E26A +13) sin3m +(-1A - 266)ca 3u] + into ☺ we get 98 [ (8 +34) Cos 3 ~ + (A-3B)s ir 3a] = exos 34 or cos3x [-ISA-26B + 9B.+27A) + sin3a [-26A +1 8B +A - 2FB Ces 3r or cor (9 A -17B) + sinan (-17A -95) = Cos 3n comparing both sides me get 9 A-17B - 17 A-9B from 17A 1 -9B H = B 1 we set putting A into 6 9 (2B) 17B 1 - 17 os (-34 -239) =17 Then from @ A. (-17 17 370 A 370 Herel P.I yt ex Casin3u-8833)

Now the general solution of the is y giren problem coft P-I x atq 6s 3x + lysinant ya) sinan 370 17 370 Now ( 3 ) y' = -32 Sir3x + 363 cas 3x ton Is to Cos 3) tex (3.9 Cos 3m + 3. 12 posinga? +3₂ Cos 3 n te (sinzul 30 cm) y (w) = -3 Casinan+ 3 Cg Ces ar t ex [ 2a sirsa - 3 sinza 2 + Ces 3 n 37 ces 3n 20 3nt a y" (r) = - Co Coszu - 9 C3 Sin3x te te [ 3.1 2 3 32 sir an yllay = – 9 4 3 - 9 Casinan te" ( 72 sing the lo cas 3n Now by the given conclitions t'c.) = 1, Y0 = 1 Mo)=2 Now putting x =o 10 7(0) = set 9th 17 370 = 2 575 Y' (o) = 363 + 137 - Gth = 1 - 137 1 => 3 4

12 37 o y" (o) = copy - 9 t 19 37 2) - 9 Cai >> = 2 37 37 from 0 G. 575 370 G 2 575 370 t 575+ 20 370 Ć 119 ty 585 340 74 Now putting a, a, into 8 We get giren problem the requined Schution of the Hry = p o t ( 34 ) as 3x + 11 sinza + te (zzo sinza - 17 370 Cos 3 u (Aus)