Question

Calculate the Laplace transform of the following time functions by applying the Laplace transform properties:

f) f(t) = 3t cos(t) g) f(t) = 3t sin(3t) h) f(t) = 2te*** – 3t sin(t) i) f(t) = t sin(3t) + 2t cos(t) j) f(t) = 5sin(t)/(3t)

Answer 1

f) f (t) = 3t Cost Laplace transform {f(t)} = { 3t Costs = 3 L{t costly = 3. S²_12 S² (5² 272 = 3. costly = 3(5²1) (8²71)2 (8²7132 | where Left Cos(atsy- S-a? (5²+0 ²72 185 9) flt) = 3t Sin3t) L{f(t)} = {3t sin (36)} =32&t sin (3+)} = 3.2(3) (s) loser {$"782)2 (:44t sinat je zas ) | 4st sihatja b) f(x)= 2te 3t - at sint 44ct) 4( 4test at sint') - 344t23} -344 tsint} $ (8-1-3)* - 3. (2442) Mato it simt Let sinat). Las 65 LG F45 (49CD Scanned with CamScanner

i fltl= t sin(3t) + 2t cost {f(t)) = ft sin (3x)} + {2x cost) = tft sin st} +244t costs = 2(3)(s) (5²+32) (3²7122 (5²+32 2 + 2. S²,2 6 S L{f(t)}=FCs) = + 2 (S-1) (5²71² j) flt) = 5 Sin (H 4ffle))= 4 5 Sint } - {to sinty as we know 48.467)e Flus.de du (here Flu) = Yata! 36 {sinth= '/s ) e ton" (o) - ştan's = I - too's 44(4) 5 = Fes) = f ( tas's] Is Scanned with CamScanner