Tutorial 4: Laplace Transform 1. Using the definition of the Laplace transform. obtain the Laplace transforms of the following functions and state the region of convergence in each case. (a) t2 (b)3+t 2. Using the definition of the Laplace transform. obtain the Laplace transforms of the following functions and state the region of convergence in each case. (a) ['94 (b) cosh 2t et+e_t [use cosh t = ) 3. Using the Laplace transform table, find the Laplace transforms of the following functions and state the region of convergence in each case. (a) 7172 2 sin 31? (b) 5.9\" + 3 2 cos 2t (C) tze4t (d) tsin 3t 4. Determine L_1{F(s)} when [7(5) is given as below. 1 (a) m 25+6 32+4 (b) 1 (C) (s + 3)(s + 7) 5. Determine L-1{F(s) } when F(s) is given as below. S " (s - 1) ($2 + 2s + 2) 5s - 7 (b) (s + 3 ) (s2 + 2) 36 (c) ( s) ($2 + 1 ) ($2 + 9 ) Answers 1 (a) 2 53 , Re(s) > 0, 3 1 53 , Re( s) > 0 2 1 (a) S (s + 1)2 , Re(s) > -1, (b) -2 4, Re(s) > 2 3 (a) 42 6 $2 + 9, Re(s) > 0, (b) - 5 3 2s s+2 , Re(s ) > 0 , 2 $2 + 4' (c (s + 4)3, Re(s) > -4, (d) 6s "( $2 + 9) 2 , Re (s ) > 0 4 1 1 (a) 16 64 Sin 4t, (b) 2 cos 2t + 3 sin 2t, (c) [e c ) = [e - 3t - e-7+ ] 5 1 Izet -zet(cost - 3 sint), (b) - 2e-3t + 2 cos Vat - 2 sir sin V2t , (C) 4 - cost + = cos 3t 2