Question

Solve the initial value problem below using the method of Laplace transforms. w" - 6' + 9w = 27t +63, w( - 1) = 3, w'(-1) = -1 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. w(t) = (Type an exact answer in terms of e.)

Answer 1

Joli Given 163 = 2 = ه - اما _ " ما 0 Taring ;ا ا 6 ادا ۹+ ({43L + +اه اها "ها ا ا ا د ا کر رہا (9) - وما 13 ( +۹l 3 = 1 + 27 27 + (ع) + كه = ما(9+ 64 م) (= + 63 [Taking w100=4, wilds=(2, لا = ا ا ا ها و + h الك (= کولاک و+م 3 6+4 و - | * ( 6 +1) 2

as Now [13 م ا م و (3) 2">= لار 3-3 - C [ 34 33 34 at ري)۴} که ce 3 a e خره -12F :1 را 3 + 34 ] ** و (+3 + 3ع ہو یا 44 - يا ] 3 G e - 6 +4 - 11 { و6 - هر 3-1) ا { (ع) - ا ء 3t e 34 -ع) 2) te het 21 +6 38 (9+) ام 114632 (1-32" A B ( ماد + + ہے - "ر3-1) (say

=) segt tz - الا (۷- کے مکرم 2 (2-4 کا لاہ(( کے) 3 tos =AS(176879) +B (6-6879 te(1²38) toon = (A +e) 8+1-6A +1-36 +D)5 +(9A-6B) A+(9B) 3 Then, Atc= -6A+B-3CtD=0 9A -6B = 63 27 9 B = D = 24 solving A = B- 3 ,C= -9, Putting there 5 and then taking inverne L-T, in J(5-6819 Ľ 3 27+631 >163t = 37-90 8-3 +244-13 3-3° -9 +3t-qe" +24te 3t 3t (6

97 Now in taking inverna L-1, and using 3 we get 3,6 6 3t with = a e witte 423+(1+3+1 +62-667te36 +9+3t-ge qe 3t +24te 7 3+ (97+6) +62-64) €34(3t+) + 3-27 234 +24(3t+vest 8 Given W(+1)=3, w(-1)=-1 ind; w=3 , want at t=1 ... From and ®,

2 - 3 се- 2 с — пе za é 3 + 64 - С) е-3 +4 -qe-3_24 е-2 + 3 - 27 е-3 – чес- 3 rin <a 2 е* 49, с.= це?+ 3 6 - | - - 2 3 То, C2

Putting there in wit= (2 e²+9) est (1+3+) clé+51) te+9+3+ 3 3t q e +24t e 3t 3t - 311tt) 3617t) 2 e tote tge 36 3 (tt) e te +27 steet st +9+3t-aft taytest 는 311tt) 3litty 2 e t ste +9+36 with or (2+5t) e 3617 t) + 9 + 3t (rea. Join