Question

please answer all 4 i really need it. its greatly appreciated

Les ) be a function on 10,6). The Laplace transform of t is the function F defined by the integral F(0) - Sa-mat. Use this definition to determina the Laplace transform of the following function. -- The Laplace transform of F®-(Type an expression using s as the variable.) It is defined for >(Type an integer or a fraction.)

Answer 1

f(t) =-843 Criven that? Let f(t) be a function on 10,00). The laplace than form of 't' is the function & defined by the integral F(s) = Sest fct) de [est cote de > L {f(t)} = f(5) = t3 dt. 3 - 8 62-56 e dt 3 est -fe 3t2.st 8 I le (-5) 75) CO3 ESCO) + + 3/ testat - 8 [ -5. (-51 560) (0013 ( 0 Co. 3 + e = 0 -5 2 + 24 [ 5 cesty est t 620 + -5 t(c)? [ 256com] feestele fisso efestet - (c) Esra! { $(0) + f ( est 10 ako Cesa)_20]] (+) 360dt : Ple) [96dt = [flow (gemise -24 5. -48 5(53). 48 t{ -54%} $4.

ji) fltliteit F(s) - 48 64 . :. It ý defined for ss lo - (547) dt FCs) = ſtéttestata e [telstilt t - (st1)t (-6+7)3) Jo -6+7 O -(5+1) 60 0 + e -(5+7) ll е (5+7) -(5+7) t (-(5+7)2) [ + (0) -(5+7) (5+7)? list (5+712 ė 6+1)(60). it 5+7)>0 5>-7 - 0 ( (5+7)? F(s) (5+7 'It į defined for/s>-7

(iii) f(t) : e36 cos st. F(S) - est (ezt cos st) dt و 1926.a)t cosst at that we know a²762 8 (a cosbt + b sinbe] cat Seatcosbt dt = -(5-3) (5-3) cosst essinst e (5-3) CosStdt (5-3) 715) a) cost wanita bungen, wie este es sim - (°C- (-3) cos(o)+ ssinco -(63).cos() +5 sin (60) (-3)1 +0 (5-3)752 1. (5-3)00:0 e if (8-3) > 0 5>-3 (16-3) - (5-3)752 F(s) = 1_(5-3) (6-3)752 it is defined for s>-3

(iv) f(t)- etcosst + ett_ 1. {{1} - Ś. from tables l{ {at} = sta s-a L { at costit} (5-0) 76² I + 1 S-] S. 1 5+1 (5 +1) 725 1{etcosst tett_ 1}