Question

I Discuss how the methods of undetermined coefficients and variation of parameter can be combined to solve the given differential equation. Carry out your ideas. (i) 3y"-6y'+30y-15 sin x + e* tan 3x Page 1 of 4

Answer 1

We know of that if y, and. Yo are particular solution doftly" + Q1 (8)y'+ Qg (1)y=-F16) and Qo (2) y' + Gi(x) y' + aglaly: BF g (2) respectively then ayı + BY 2 is a particular solution of Qorxy" + a, (a) y' + ag Gys a F1601 + F2(x) Considen - 0 - 34" - 6 x + 3 x = 1s Sinx + cº lan 3x . The corresponding homogenous equation is 3y" - by' + 30y=0 The Auxilliary equation i 3m-6m +36= 0 7 m 2m+ 1020 - ma 2 + 54-40 Thus, = 2161 - 1131 de = @" (e, sip3x + Cg Cos 3x) is the compliementary solution of o . Consider 3y" - by' + 30y = N Sinx - o the Non- homogenous term is sinx

we shall use method of undetermined Cofficient to find the particular solution of @ The UC set of sine u s=î sinx, Cos23 clearly yes y does not belongs to s lo 67 ya A Sinat B Cose 4b = A Costa Bsinx then You = -A Sinan B losse Thus 34" - by' + 30y= sind implies that - Sina - 3 A Snx - 3 8 (osx - 6 A (63x + 8 Sinx + 34 8 Sinx + 3 Box 27 A +682 - 3 A +63 +2012 I - 6 A +27B=0 -3B -6A +30 B20 A = BP and Arts 765

7 Ypa is sinx + 7 Cose i particular solution Consider 3y" - by' + 30y = e" tanza -o Here we shall use Variation of parameter to find the particular solution of © Single y; = e Sinse, ya = és 3x are two linearly independent solution of corresponding homogenous equation of o us, ypa bresin 2014 by los 30 where we, and Ug to be determined

W Gala D F(*) y? J ao w[Y' ya'] . .. l.. Wiel:.3024) es tan 3x e Cos 3x de 3.-3ed2 1053X sin 3x dx = | : 4 (sin3xdt: 3 :33 and Vack) = U (K) f Fla) yo (3) dx = s iulan 3.-3 e27 J do w

2 1) sin 31 COS 306 da * (sec 34 dx + 45 (0:24 de - tin (sec 324 tande) + Singa In ( Sec 214 tan 30) + sin 3x Yba - Cos3x. et sinde - in (sec34+ tan 31 ) e "(0532 s 37 I er sinse 6s 32 i particular Bole of ® hono finally, ya et sin 34+ Cg (0534) + 154 Sinx + 15:27 Cose La Cos 34 e*sin 3x -ta in (sec 3x + tan 34 ) e46s 32 trebne sin 34 61 32 is the general solution of o