Question

Solve the given initial value problem by undetermined coefficients (annihilator approach).

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

y(0) = 2

y'(0) = y''(0) =1

Answer 1

Contact It (ka) to ne - Cole opt(x2) Ogo Ausiliary equation is m3 +9mzo Given initial value problem yay Cos (3x) ; y(0) =2, y) =y" o =) the general Solution coill be the sum of the Complementary solution and particular suktion. first we find complementary solutions by ylltay? - mm (9)=0 Ore m=0, m +9=0 mao, m= £i therefore, the complementary solution i (riget (r), ľ=( 9 CCW = G + C₂ Cos (3x) + C₂ Sin (3x) Now, determine the particular solution to y"4aya elaska) by the method of undetermined Coefficients. NOW, the particular solution of the form: CHED Elyplats ared Cos (34) +a, e ash (3x) → Up (a) = aie? Cl (3x) ~39, e? sin 131) +892 ed sh(58) + 3a2 et Cullsx) # yawa-260129 Gay (31) 4189,095m (82) - 1892e? Cop (34) ~ 26a2e? Sch (3x) Now, substitute those values i yt24'= e($(3 x) => -2691e61132) 418a18?sh (3x) -18 9, Pyrex -2602ed sin (3x) + 9 (a , e? C (3x)-3a,e? Sh (34) + Base Cas(3x) 49,0?sh(3x)) =e964 (33) (-190, 4902) ex Cos (34) + (- 99, -1792) Tshish aea (30) Equate the coefficients of e a C1 (32), e shisx) on both sides, ure او او او . ora

are get a = - 17 op therefore 370 substitute a, and az ihto yplassere Carlsx) +age shia yllen) =-9 Caco (Ba) -9 (35h (sa) the Cy (31) -17a, 49 a 21 o'tqaj -1702 = 0 to solve thes, & are get a = 370 A2 = 9 6- Plyte with notcandan 37:01 therefore a Yp(a) = -17 e Glsx) + 2 ea sinten How consider initial Conditions the general solution is y(a)a9 cca) + Spla) CHAOS + y(a) a 41+ ca Gar(3x)+ Cg Sh(34) 2talo to born 17 & Cul (3x) + 2 etsinler Contact 370 (x2) Na tay'la) -3C2 546 (33)+362 Cos (3x) + 5 Calderon (een the Tsh (31) ( 9 Corona e 370 e&G (31) + 3 exsh (30) 37 370 (x 2 now, ylo) = yco) = ato turn 52-7009 soos in die syco) 3Cz57-1*= 12/37 (2322 Fins - (x2) gluco) e - 9 C2 + 14 =12 C2=-2/37 6.2010 by (1), = 21/0. (o substitute these values into yin). y(a) = 34 C1 (58) + sch(3x) - 570 % (sv RI 10 Cell 37 91370 ea "Sch(32)

reb Simiply: yena stol-C172 420) Cay (32) +3(34?440) Sm(3x) +777) c
if you have any doubt in any step please comment