Question

Using the Tables and other properties, find the following Laplace transforms. State the number of the formula you use. a) L{ (t? +4)e 6+ - cost } b) Use L{cºf(t) } = (-1), da F to find L {t sinh 5t) } (yes that is the hyperbolic sine function, sinh ds"

Answer 1

Definition: let that the flt) be function domain of FCE) a function of varjable t such est fet) is integrable on to,oo] of values The laplace ot denoted by the integral S. for some transforma L {f(t)} = F(s) est FCE) at Thm - It Fict) and fact) are two gunctions of t and a and a Contants then {q filt) & fact)} Lofo(t)} + Co ?{fa (t)} are any rewo

Laplace transform of some elementsaly functione [{f(!)? f( O fact) + Cofa (E) GFCS) + C2 Fe(s) 40 f(!) (-49" dh F(s) ds" fCt) S f(s) ds pat sa S-a 1 - S sin hat a S²92 s> lal Cas hat S Sla) s²_a2 sinal s2ta² + sin hat Qas (52 92)2 t coshal g2 + a2 (5²-02)2

solution : Let F(t) = (22+4) est cos 7 itfct)y=1 (+2+4)ert cos 777 using theorem 1&f(t)} = 18 (22+4)eat. cos 747 = 17212+4)067- Ly cost as LY F(t)) F (s) 1 F(s)LV t?eft +4 est} -18COS 7E} first we find LY Pbl 44 got 1X12 6 + 4 got L {p? &t} + 4 LX est 2) Use, LYt" f(t)} = (-1)d"F(S) dan be the abore find using Can :: Lat? est? Dormula bete f(t)- SE :. FCs) = 18F(+)7 = bettt (Yeah. S-6 S-a ther. 1412 Sty = (-1) ² d 2 F (5) (² d² CA ds asest LTE est}= de a

। 212 d 4 । (5-6)2 (4 (क) का ds :12 GET 2 (S-6)3 also, [रहा 1 1 5-6 . (همان) S-a wsing eq? 8 and 6 q? @ becomes 1210 +4 ENT (S-6) + 4 S-6 How to find [ Rasa Le cos at? 42+52 as S 570 । र cost) : S S 520 212 (2149 [2cos 52149 संबर using eq" o and 6, egr ① be comes F(s): S + (-6)3 4 S-6 s2449 1122+4)ी - cost (56)3 (s-6) 3149

6) salution : To find LY +(sin h st)=18+ f(t)? : f(t) = sinhst . wing F(S) = XF(t)? La sina st} L & sin hat} = a s2-22 sslal : F(s) = 12 sinnst} = Tas sy is 52-25 wing formula 1 X " FCH)} = (-40° do F(s) dsh : 12t sinnst} = (-1) d F(S) as ds -25 (1) sd st =(-1)(-5) (32-21) (3-2)2 dr 1 1. 4(x2) (x-a21 (28) (52_255 10 S ALYt sinnst? (62-21)2