問題1.1 第一移動定理の応用 ラプラス逆変換

第一移動定理
L(exp(at) f(t)) = F(s-a)
L(exp(at) cos ωt) = (s-a)/{(s-a)^2 + ω^2}
L(exp(at) sin ωt) = ω/{(s-a)^2 + ω^2}
L(exp(at) t^n) = n!/(s-a)^(n+1)

35. 1/(s+1)^2
1/(s+1)^2
= L(exp(-t) t^1)
= L(t exp(-t))

36. 12/(s-3)^4
12/(s-3)^4
= 2 * 3!/(s-3)^4
= 2 L(exp(3t) t^3)
= L(2 t^3 exp(3t))

37. 3/(s^2 + 6s + 18)
3/(s^2 + 6s + 18)
=3/{(s+3)^2 +3^2}
= L{exp(-3t) sin 3t}

38. 4/{s^2 - 2s - 3}
4/{s^2 - 2s - 3}
= 4/(s-3)(s+1)
=1/(s-3) - 1/(s+1)
= L(exp(3t)) - L(exp(-t))
= L{exp(3t) - exp(-t)}

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

40. 2/{s^2 + s + 1/2}
2/{s^2 + s + 1/2}
= 2/{(s+1/2)^2 + 1/2^2}
= 8 (1/2) / {(s+1/2)^2 + (1/2)^2}
= 8 L{exp(-t/2) sin(t/2)}
= L{8exp(-t/2) sin(t/2)}