Question

Determine the inverse Laplace transform of the function below. Se -45 s2 + 10s + 50 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. 4s Se 2-1 s2 (t) = + 10s + 50 (Use parentheses to clearly denote the argument of each function.)

Answer 1

'{ se48 Question : (1) gatlost 50 Solution : Flow, the propeaties of inverse Laplace transform, we have li Fiest shifting beoberty: If I'f Fis)3 = f(t), then š's F15-013= et f(t) = eat I' F121) .: I'{ F13-a13= eat I's Fr%)] -(1) lii) Second shifting beoborty : If a { flsi}= f(t), then we have F131) = f(t-a).Ult-a) —42) where Ult-a) is unit step function defined by Ult-a) = tca S Jet 105556 s Note Set 2*583 725725 Tes es ē as I tsa Now let FIS) =

12) 5 .. F(9) s Sa+245x 9 +53 +25 s ( 37 sjá ta5 uf:(a+bhe (= CPF296+6) st5-5 ( Notel 1-375)2+25 (365) 1st 512425 (3+594+25 F: 13+5) - F (965) Fil-3+5) - F 13+5) -(3) where st5 Fil3751 +4) 1st sta5 .: Fot3) = atas 45] and Fel 3+5) = -16) 6975) tos - Fa (9) = ftans 47 Now taking the inverse Laplace tlausform of equation (3) , we have

(8) d' Fea1] = I'ç Fi(3+5).–F 1945) { F1(8+51) – I'{ FE(3+5) Il Pan Fi(+ 92 Ft-3)? ai mo'p Fi(2+ abati"F49)), by linearity property of inverse Laplace transform where all and as are constants - 718) JF, 13+51) – J'( E(3+5} = 27.12) - L' (23] by first shifting beobeety given by equality at 21:1313 - J' E) 2** [ 7'3) - JPEG 89425 iſ by equatlong 159 and (7) pale au regte ty stolets yven i

14) = =** [ zil #583 - 2375] ēst [ Cos 15t) – Sin(58)] I'l 9 249* Coslat) and om te gaan? = Si (at). d's F1933 = ēst ests Collst) - Siv (58) of f1t) = at Cos(56) – 3M (57) PRI is we have Now we have F getlos+50 gatlost 50 I's 349 F131), where FO) 827108+50 fft-4). Uſt-4) , S by Second Shifting property given by I equation (2

sees Setlost56 १) .:' - flt-4) - U(t-4) Now from equation (8), we have $(t=4) = otty Cas5(+-4) - Sin5t-4) - Fit-y) = 2 -) { ces( st–2,9 – SM (56 -2 from equations (9) and flo), we -110) have وام و stlost 50 (2-4) of als seus 1327105750?re) (5+-20). 1(4-4) 572-93{co9[57-209) – 9.4 ettste) { Co951t+4) – 31,5(7-4)), +34 = f bijis = tch 1930(7-4) =

() Hence we have 8 ?비 Xt09+50 넚 SIE-) } cas(ste-- 3.5+ (s(e-i), t >> = ,t<4 오 회 95 Sat) 왱 E-M) Co(s(==)) - Sp(s(E-) 3. Ult-4) = Whene U(E-4) EN ) 이때