Prove:

(a) Commutativity, f * g = g * f
(b) Associativity, ( f * g) * v = f * (g * v)
(c) Distributivity, f * (g₁ + g₂) = f * g₁ + f * g₂
(d) Dirac€™s delta. Derive the sifting formula (4) in Sec. 6.4 by using f_k with α = 0 [(1), Sec. 6.4] and applying the mean value theorem for integrals.

(e) Unspecified driving force. Show that forced vibrations governed by

with ω ‰  0 and an unspecified driving force r(t) can be written in convolution form,

y" + wy = r(1), y(0) = K1, y'(0) = K2 1 sin wt * r(t) + K, cos wt + K2 sin ot.