Question

Use the Laplace transform to solve the initial value problem: y" - 3y' + 2y = 4t + ezt, y(0) = 1, y'(0) = -1

Answer 1

Salection : The given initial value problem is , - D y" – 3y' + 2y = 47+e31 yo)=1; y' (6)= -1 | with A second order - linear differential equation of the form yll of pylt qy' = Rct) can be solved by using laplace transform and the inverse Laplace transform. The Laplace transform of a function f(t) is F (S) which is denoted as A n al LXfCŁ)= F(S) . Suppose [ayct)} = Y(s), then the Laplace transform of derivatives is given by following way: 147= y(s) — ☺ y= st(s) - Yco) — ☺ J y'? = say(s) - Syco) - y'Co) —© 11 g = côY Cs) - s296) - Su! cs)- g" (o)-S) To solve our giver diff. ego , yll - 34' + 2y = 40+ €315 Laplace transform on both Take - Lay" - By'+ 2y} = {{4E4 e3t] Lylly-3 294"}+2 29 y}= 4 LYŁY + LTesty

Using eqo , ③. and ③, and also LYtY = 1 and Ideat? = we have, SPY(5) - Syco) - y'co)-34 3 Yis)- yCo)} + 2 y(s) = 4(1 + 1 S-3 . using YC) = 1, yl(o)= -1 s27(s) - 5+1-3457(s)-1} +2 Y(s) = y + s2fis) - 5 +- 35 F18) + 3 + 2 y(s) = 4+ 2 S-3 (33-35+23 768) -5+4 4 + 1 (5² 35+2) res) = 4 1 ts - 4 S- 2 968) 302_342) + (63)(8-3:42) TC-2842) Gesse) ris) = 46-8) + s2 + sø(5-3) -452(5-3) s2 (5-3) (5²-35+2) Trs) - 54-783 + 1352 + 45-12 52 (5-3) (82 - 35+2)

s4 - +53 +13'52 +45 - 12_(Fachoning denaginator) s. 52 (5-3) CS-2) (S-1) factoring denominator we can write Y is) as, Tes) = 84-7 53 +1352 +us- 12 S2 (8-3) (S-2)(5-1) td The form of partial decomposition is, est 84_753 +135²+45-12 8² (5-3) (5-2) (S-1) = A + B ats-3 s-2 = AS(8-3) CS=2) Cs-1)+8(9-3) (-2)8-1) + C(92) (CS-3)(S-2) +0S2CS-2)S-1)+. E. (52) (8-3) (5-1) . 32 (-3) CS-2) (S-1) The denominator are equal, so the quantity of Demerator. 84–183 +1352 +45-12 = 54 (8-3)(S-2) C+52 (5-3) CS-1) E +2C5-2)(3-1) D+ s(-3) (5-2)CS-) A +(5-3) ( -2) (5-1) B . 54783+13 S2 +45-12 - S4A + 33 c + 84 D + 54 € -693 A + 338 -5330 - 353 D - 453E +1152A -652B +692 +252D + 382 € -6S A +11 SB - 68

collect the like terms s'-753 +13.52 +45-12 3 S4 CATC+o+ E) +33(-6A +B-57-3D -46) +52 (11A - 62 +6C +20+36) +s(-6A +11B) -(68) the coefficient comparing of like terms in both sides a 1 = Atc+DTE -7 = -6A +B-5C - 3D-40 - 0 13 = 11A - 6B +60 +20 +3E-5 *4 = -6A +11B co -12 = -6B - as -12 = -6B JB-2 From , 4 = -6A+ 11(2) = -6A +22 4 --GA +22 -18 = -6 A LA = 3

f Sawing all this A-3, B2 systems we have in Cari Dağ E = -2 .: from 299 2 | 54-733 + 1852 +45-12 = 3+ S2 (S-3) (S-2) (S-1) + $-3 3-2 Hence from ean eta Yes)= F song . Now taking sides. inverse Laplace transform on both *P799)3 3 '{4}+3714714 4447 2 af 11 Ts-2 T I using formula, il 17 2 - 2 we have yt) = 3 + 2t #1 1. 2 . 2 : - y08) = :3 +2+ + 1 € + t - 235 Hence, the general salution is , 3ct) = 3+27 - Det-1 et +4 037 Ict) = 3+2-2 -le