Question

Find a particular solution to the following equations

Problem 1: Find a particular solution to the following equations +y= t + sint + 1let + Již + y = 2t sint dy-4d4 = e2t + + +5+ + 4 dt5

Answer 1

사2 The given differential equation is dy + y = ť+ sinttllet is. (03+) y = ²+ sinttllet Particular solution Jp = 1 (+²+ sint illet) where D = d. = (t) 1 D3+1 + sint DHI + 11.1 D3+1 at at :Dt. 1+1 D) Fla) = (1+D3) (t) + 2 sint + 11. I et { het Leet = (1 - 3 + (03) 2 ---) (+3) + 1 12 sint + Let : 1 Sin at I final TE(D²) FEB =(t²-D? (+3) + Do (+3) .--) + b sint + let (-12) D+I DTI = (x3 - 6to) + 1 + 1 +D sint & l et = t3-6 + (!+D) sint + Let = +²6+ (1+D) sint + het = (3-6 +(! + D) sint + het = +² 6+ 2 sint + & cost + het :: p = +² 6+2 sint + { cost + het ] I-(-12)

& The given differential equation is de 2 + y = 2t sint ie. (D²+1)y=2tsint Particular solution: Yp = a (atsint) = 12 (2t & imaginary part of eit) al = imaginary part of a (atelt) =imaginary part of ze it I t at Fine a FLD ta) t otiz itl I Fis) e Plt) zeet D²+2iD T: +1=0 ? zeit + imaginary part of Rett 12 +212 + 12 (4) imaginary part of zeit 1 (t) { : 1² - imaginary part of zett 2iD (2 + 4) =imaginary part of eth (1+225 it) =imaginary part of elt (- 2 + 2 ...) (+) imaginary part of eit (+- 1 to urn) =imaginary part of eit ( + - 1 (3} {&t=State = 12 imaginary part of ett 1 2 - 2 / 7 =imaginany part of cost t isint (1 + it ] {:- 1 - 1 - i J = 7 -2 cost + 2 sint

③ The given differential equation is e - 4 edt +++54+4 ic. (05-403) y = (2+ + 8 + 5++4 Particular solution: Joosenz (eze+t+st+4) Co gor ' tip tig3 (+5++4) ska jest boll- big" (8+st+4) Sorota S55 e" cus' 6 Due to the litt )6t+si+yl 5 best 42 (42+31+4)** 04+75+25)+ to *(4+54)+ - Lezt ++4+ 4 (2+0+0) +0+.- - get +54 +4+1] = 1 2 2 2 2 2 2 1 13 ( 7+ 5++/ ) ( 1 3 (²+1+2 - SSJ (+²+51 +23 de? as (1 + 5 + + 27 ) de)? -Seat Sf to y de + hli +