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10. (a) Use the Tables of Fourier transforms, along with the operational theorems, to find the inverse Fourier transform of 40 w2 - 13iw (b) Use Fourier transforms to solve dy -5t + 8y — 9е эH (). dt 15t H(t) 1 8t (а) (1) Н, 9 Then solve for 5iw (b) Apply the Formula of transform of derivatives to get: (iw+8)Y(w) Y (w) and take the inverse transform to have -8t у(0) — Зе 5 н(t) — Зе 8 Н((). -

Answer 1

oFingt we State a TF Propenty wnene aYo atiw then the Founier Tnvense of thig F(W) i -at HC), atiw whene Ht) \a the unit Step function So Lel 8ius-5i 0 i3 w(w-1-5i (w-81) w- w-8i w-81 F() = +iw) Novw Takin Pounier Inverde b0th Side we havo 5tiw

-8t |b7 usin tne Pno Perty -5t H (t) Auswen This is the heowined invenge. Finstwe State the Pno perty of Mne Founien thansporm o a derivative of a ?unction So whene Yw) is tne Fouhier thans POrm of t), Given ditterential evation i3 -5t 8= 0e ) dt takin Founier thanS Ponm b0 th side we have dt bY he (11 an d Proferty 1 jw Y(w) t8Y(w) 5tio ((3) 811u(5ti) 5tiw

Takin Founier Tnvense both-Siae we have - 8 tiw tiw - -5t = 3.e -8t MA) by the PnoPenty 3e