Question

Please do not omit any steps.

Use the method of Undetermined Coefficients to solve y" – 2y = 8 sin’a; y(0) = 0,7(0) = 0.

Answer 1

1 - 4 cosau. m - 2m = 0 y" - 2y! = 8 sch²x. y - 2y = 84 ½ - 1 cos2x) 9"-2y' = 4 Auxillary equation of mim- 2) = 0 complementary solution Ye (x) = e, toze 012 271 let yp (X) = Axt Bsihzat ccos2u Yp '(x) = A + 2B cosan 20 sthaa ye - 4 B shax - 40 cosaa. put these values (sh U - 4 Bschex – 4 Ccos 2x ZA - 43 cosent 4c sih2a= 4 - 4 cos 2x. Comparing the coefficients to - 2A-4 ere - 4 B + 40 = 0 - 4c- 4B = - 4 I A = -2 B=0 CL 18 ) = – 4 - = 2 2 1 = 1 an 4 shax + cos2x + 1 an Yp (x) = 2

& y(x) = Ye(x) typ (x) 22 4x) - cit cze 2a + schaat cola 2 y (0%=0 citoz ta [C, + c ₂ = - 1 / 1 y'(x) = gege 2x 2 + cos2x - sinza y '(o)= 2Cg - 2+1 2 cq-1=0 Heza! 2 1 C1 - 1 za 2x+ sihaa cosan y (x)=-1te V 2 2 2