Question

(1 point) Use the Laplace transform to solve the following initial value problem: y" + 3y = 0 y(0) = -1, y(0) = 7 First, using Y for the Laplace transform of y(t), i.e.. Y = C{y(t)} find the equation you get by taking the Laplace transform of the differential equation = 0 Now solve for Y (8) and write the above answer in its partial fraction decomposition, Y(s) Y(8) = B b where a <b sta !! Now by inverting the transform, find y(t) =

Answer 1

911 + 3y.co ادا ) y loan, y'roza. we know that. ya shuis) - 5.4(0) - 4'10) ylsy(s) - 910) alven, Log in ou 4 (0) = -1 y(0) = 7 ylla sky is) - 5910) - 6'20) 410): -1;; 4!10)-7 y = 5913) - S/-1) 7 "la guls) + 5-7 NOW in s sly(s)) - 410) 41= 5(yrs))-1-1) - Syls)+1 -3 NOW y+3y! 20 from , ③ ;', y valus (sky19) +5->) +3/34/3) +1= 0 Jlsty(s) +5 - ) + 3 (5915) #)| 20

shy(s) ts -7 +354(3)+ 3 = 0 15 +35) y(s) +5-4 (sh+3s) y lo y (5) = - (5-4) 4(3):1-5+4 32+35. NOW 413) - S +4 stis By partial de composition - 5+4 s2+33

-574 s(s+3) A S + st3. : -5+44 = 'Als+3) + 813) 8573) Sist) - 5+4 ASTA3+ BS -$+45 (A+B). + A3 By comparing 's' By comparing constant -AT1 4 AB NOW A-4 (from 4.) -1 4 + 3 -1-42 B-3-4 3 BE -st4 St3 S(+3) Aa وربما olf + $(-) 3 (5+3) c

S+3 . 4 3S (s+3) · (1) - T e3 use * - yle) (SA) 놀3출늠3 Apply innelse iaplace transtorns 1. 3 4 '의 Me) Asus 36 YE) [4 Answew 0 (Grous) +-) +3 ssts) +1 St나 y(S) s3s (S) a + 3 S. + 3(St.3)