問題1.2 積分による新しい逆変換

13. 1/(s^2 + 4s)
1/(s^2 + 4s)
= 1/s(s+4)
= 1/4 {1/s - 1/(s+4)}
= 1/4 *1/s - 1/4 * 1/(s+4)
= 1/4 L(1) - 1/4 L(exp(-4t)
= L(1/4 (1 - exp(-4t))

14. 4/(s^3 - 2s^2)
4/(s^3 - 2s^2)
= 4/s^2(s-2)
= 2/s(s-2) - 2/s^2}
= 1/(s-2) - 1/s - 1/s^2
= L(exp(2t)) - L(1) - L(t)
= L(exp(2t) - 1 - t)

15. 1/s(s^2 + ω^2)
1/s(s^2 + ω^2)
= 1/ω^2 (1/s - s/(s^2 + ω^2))
= 1/ω^2 * 1/s - 1/ω^2 * s/(s^2 + ω^2)
= 1/ω^2 L(1) - 1/ω^2 *L(cos ωt)
= L(1/ω^2 - 1/ω^2 cos ωt)
= L{1/ω^2 (1 - cos ωt)}

16. 1/(s^5 + s^3)
= 1/s^3(s^2 + 1)
= 1/s^3 - 1/s(s^2 + 1)
= 1/s^3 - 1/s + s/(s^2 + 1)
= 1/2 L(t^2) - L(1) + L(cos t)
= L(t^2/2 - 1 + cos t)

17. 1/(s^3 - s)
= 1/s(s^2 - 1)
= s/(s^2 - 1) - 1/s
= L(cosh t) - L(1)
= L(cosh t - 1)

18. 1/s^2 * (s-1)/(s+1)
= 2/s(s+1) - 1/s^2
= 2{1/s - 1/(s+1)} - 1/s^2
= 2L(1) - 2L(exp(-t)) - L(t)
= L(2 - 2exp(-t) - t)

19. 9/s^2 * (s+1)/(s^2 + 9)
= 9s/s^2(s^2 + 9) + 9/s^2(s^2 + 9)
= s{1/s^2 - 1/(s^2 + 9)} + {1/s^2 - 1/(s^2 + 9)}
= 1/s - s/(s^2 + 3^2) + 1/s^2 - 1/3 * 3/(s^2 + 3^2)
= L(1) - L(cos 3t) + L(t) - 1/3 L(sin 3t)
= L(1 - cos 3t + t - 1/3 sin 3t)
= L(1 + t - cos 3t - 1/3 sin 3t)