Question

ملاحظة: اجب على سؤال واحد فقط Q5: prove that the L{ear} 3 = 6-a) by definition of 1 f Laplace transform ?

Answer 1

Ans. Prove (fear} auta Laplace transform definition says that L{f(t) -st e f(t) at F (S) - ax Now Heue the function is given e and hele let net so the function becomes eat Now put the function in equation 0 so we get f (t) dt Lino Jesus F(8) - Ten.com os (14)- eu? { Now solve jest eat at

jest, eat at Now this can be written al ->) - s eat-st dt 2 Now let U= at-st. differentiate on both side with respect to t so we get du d [at-st dt dt du 0-S dt dt L du 0-S Now put all the values in equation @ =) Jeatret de ) LI S-a S eu dt a" here a=e -> =). 04 Jau du Inca) Apply exponenhal rule

at -st S-a Now apply the limits so we get 8 atast = ) e [ Q-g =) alco)-S (0) e a(o)-sco) e a-S a - S a - S Note : a (or) – sloo) e o a-s SO and it is assumed that 9-S >, and es_eaxo so that the result is

*[-(+) [: =) a-S = ) ㅗ S-a f(s) Ans. Hence Proved.