1)

y(0) =0

y' + 6 y+9 ∫[τ_de_0_a_t] (y(τ) dτ) = 1

Entonces

s F(s) + 6 F(s) + 9 (F(s)/s) = 1

(s^2 + 6 s + 9) F(s) / s = 1

F(s) = 1/( s(s + 3)^2)

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Ahora suponemos

1/( s(s + 3)^2) = (A/s) + ((B(s + 3) +C)/(s + 3)^2)

1/( s(s + 3)^2) = (A(s + 3)^2 + B s(s + 3) + Cs)/ (s (s + 3)^2)

1/( s(s + 3)^2) = ((A+B)s^2 + (6 A+ 3 B + C) s + 9 A)/ (s (s + 3)^2)

Identificamos

A + B = 0

6 A + 3 B + C = 0

9 A = 1

A = 1/9 ; B = -1/9

6 (1/9) + 3(-1/9) + C = 0

C = -3/9

Queda

F(s) = 1/( s(s + 3)^2) = (1/9) (1/s) - (1/9) (( (s + 3) + 3)/(s + 3)^2)

ó

F(s) = 1/( s(s + 3)^2) = (1/9) (1/s) - (1/9) (1/(s + 3)) - (1/3)(1/(s + 3)^2)

f(t) = (1/9) - (1/9) e^(-3t) - (1/3) t e^(-3t)

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¿Porque nosotros tenemos que hacer eso si te lo dieron a vos?