Question

A. Compute C{te-2} using the Integral Definition of Laplace Transform, be sure to note the region of convergence. 52-55+2 B. Compute { (5-1) (82 -5s+6) C.Compute c-' {e*+2+1}

Answer 1

Soln (A) To find a {te 21 } mortem operation of multiplying fees bougie them integrating belon Caplace transformation ih s te-24} - e-S4 {te-243 defe t e-(572)4 dt. o o O By using integration of parts (3+2) t -(5+2) - (5+2) (st2le 1 e-s+2)+ t E (5+2) S+2 ot = o 0 -1 St2 ($+2) 1 Anss h{te-2t} = (5+2) ² is Region of Convergence Real (s) > -2

Soln (B) Given, problem. L-1 S²-55+2 (S-1)2 (5255+67 oral S²-55+2 L? } (st)? (5-3)65-817 By using partial fraction Concept + S2-5572 A B с + (S-1)2 (5-3) (5-2) (S-1) (5-12 S-3. D - S-2. S2_55+2. AS-S-2)(5-3) + $(5-745-37Fcfs-12 (S-2) + (S-1)-(3-37 Als3-652 +118-6) +B[52-55 +6) +C{s3 _us? + 55-2) + ( 33 =552475 -3) Coefficients of both side Comparing hel have.

Date Page Att D = 0 -6A + B - 40-5D=1. TIA-58.75C+ID = -5 - 6A +6B-20-30 = 2 4 Schreing for A, B, C, D. Matrix Aug mentegd. form is - 0 1 - - 1 - 11 -5 7 -3 -5 6 -2 -5 2 Apply Ray Rq +6R1 R2 R₂ +6R , R₂ R₃-11R, and Ry Ry +6 R1 1 1 1 0 > 1 1 1 2 -5 -6 -4 -6 4 3 0 1 1-5 9 O Rza R + 5 Rą and Ry Ry ble o 1 0 1 1 0 O 0 9 4 -8 1 0 -3

Date Page Apply Rza - 1 f. 4 1 0 0 1 2 O 0 O 1.1/4" 11 1. O. -3-1 -4 0 8 Apply Ryt Ry + 8R3 0 10 1 1 2 1 1 1 000 11 0 0 1/4 O 0 -1 -4. is the required reduced form which 1 LO 1 1 1 a loop Odo O A B C D 1 114 0 0 --4 Solving we live. A + C +D = 0. B +20 to ct 10 시 -4 0. D=4

Date Page B = - De IA - 1-4. - 3 1 B + 2x-1 + 4 = 1 |--2 B=-11 А -3,5 There fere. Fulling values of Aybolydin o s²-55+2 - 1 (S-1)2(5-3)(5-2) | 5-1) (S-1)2 15-32 + 4 (8-9) Applying Laplace frame form on both siche of! (2) L-S S²-55+2 2 = -3L-S -3 L-IS (S-1)(3-3)(3-2) Srls ht 1 7-2-111 2 15-112 (S-3, +4L-IS_1 2 { $2} s-2) . Now . 1. L-16 is

Date Page hal e By using first transform. fos) flt) {f(-a)} = eat ft) therefore, If ie if then e It s-l (5-12 - etxt L-1 est S-3 elt s-2 Putting values in nel have Anahtq s²-55 +2 15-12 (5²55+6)/ -zet tet est + 4 eet which is the required result

Sono io Guren, problem ht qers s? S2+25+) 08 e-s s² TS+12 148 By using formula L-131 shtil th ni hi 1 t 11 0 S2 By using first Phitling property FID {fis-a) } FIA) ie then (b eat

there fore. L-1 I is liye e-tot {from (1) } or 1-1 is 41 2 3 4 et > H (2 By using Multiplication by powers of s. property it if List(s)} = f(t) and flo ish f(s) =O then L (7) up dan from L-1 s2 d² Stet (5+112 dtal =dsat et + e-t? = -(e-t-te-t) - et S2 2 = te-t-e-t-e-t. It 11 Lalits)

Date Page or Thu S² 2 (S+ te-t-re-t property of By using second shif shifting Inverse of Caplace Te if hf (s) - Fle) then L-is e-as f(s)} = F (A-a) u (4- a) wheie ull-a) uf tra, from we have = 0 if eca 1 to 5+1 j2 L-1 { e-s x S2 H-1e-4-15 -2 2-8-)? Ang; k-$ por een ec-23$ 18-12 ult-1 or e-s s² ell-7)(4-3) UCH S²+2541