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9. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of iw 4 + 9 w2 9 w2 (b) The function f(t) satisfies the integral equation: OO -4u Н(u) du + 6sgn(t) е З, f(t) 0- ft - u) е" = 4 e -OO Find the Fourier transform of the function f(t) and hence find the solution f(t) 7 "(1-)н, (b) Transform the equation by using the convolution Theorem: F(w) 3t "Н() 6 (а) 1 = 2F(w) 12iw iw4 9w2 Then solve for F(w) and take the inverse transform to have - - 2е 3 Н (€) — 14е3" Н(—t). f(t)

Answer 1

a) we have tt find Founie invense transm 4 3 $ince Fel 2a and 4 (3 + w) (3- w) 9t 3i 3-iw 3t0 4- 4- 94w2 3-iu 31 w 3u 3 u + 3-u -3t 3t ukt) b- uSing a tiw at uL- a-Tu 4u -31t HIU) du + 4 - jwul dt -tu e fli-4) Ju flt) e jwt dt -31 31 Ju e -iwt -+f HI e e 4 Flw) Flw)

Flw 4 Flu di 4 Flw) 4 Flu) 4 w (31 fu)t KmI-8)" Fw) aiuw Flw) 4 3-iuJ 3+ru Flw) 4 Flw) 3- 4 lw) 4 tiw)Flu) 3-t u 3-iw Lw FLu) Flw) -2 64+ -12 ( 4 ju) Flu) 3t ul J utt rit) -12 S0 e Fiom pan 3t 3t ul+ 2 e