Equações diferencias ordinarias

Universidade Federal do Amap´ a
Coordena¸˜o de Matem´tica ca a Disciplina: EDO - Turma 2010, Matem´tica a Prof. Dr. Guzm´n Isla Chamilco a

Transformada de Laplace e Sistemas de EDO’s
1.- Considere-se a, b constantes. Encontre a Transformada de Laplace das fun¸oes dadas: c˜ i) f (t) = t, ii) f (t) = t2 , iii) f (t) = tn , iv) f (t) = cos(at), v) f (t) = sen(at), ix) f (t) = eat senh(bt),

vi) f (t) = eat cos(bt), x) f (t) = teat ,

vii) f (t) = eat sen(bt),

viii) f (t) = eat cosh(bt),

xi) f (t) = tsen(bt),   2 

xii) f (t) = tcosh(bt),

xiii) f (t) = tn eat   0  se 0 ≤ t < 1

se 0 ≤ t < 3 xv) f (t) =

xiv) f (t) =

−2 se t ≥ 3

t2 se t ≥ 1 3π 2

xvi) f (t) =

  t se 0 ≤ t < 2  0 se t ≥ 2

xvii) f (t) =

   0  

se 0 ≤ t <

   sent se t ≥ 3π  2

  1 se 0 ≤ t < 4     0 se 4 ≤ t < 5 xviii) f (t) =      1 se t ≥ 5   0  se t < 2

xix) f (t) =

  sent se 0 ≤ t < 2π  0 se t ≥ 2π

xx) f (t) =

xxi) f (t) =

  0 

se t < 1

(t − 2)2 se t ≥ 2

t2 − 2t + 2 se t ≥ 1

 se t < π  0     (t − π) se π ≤ t ≤ 2π xxii) f (t) =      0 se 2π ≤ t

xxiii) f (t) =

  0 

se t < 1

t2 + 2t + 2 se t ≥ 1

1

2.- Encontre a Transformada de Laplace inversa das fun¸oes dadas: c˜ i) F (s) = 3 , s2 + 4 ii) F (s) = 4 , (s − 1)2 iii) F (s) = 2 , s2 + 3s − 4 2s − 3 , s2 − 4 s2 2s − 3 , + 2s + 10

iv) F (s) = vii) F (s) =

s2

3s , −s−6 2s + 1 , − 2s + 2

v) F (s) =

s2

2s + 2 , + 2s + 5 s2

vi) F (s) =

s2

viii) F (s) =

1 − 2s , + 4s + 5

ix) F (s) =

x) F (s) =

8s2 − 4s + 12 s(s2 + 4)

3.- Esboce o grafico da fun¸ao dada no intervalo t ≥ 0 c˜ i) f (t) = u1 (t) + 2u3 (t) − 6u4 (t), ii) f (t) = (t − 3)u2 (t) − (t − 2)2u3 (t), iv) g(t) = f (t − 3)u3 (t); onde f (t) = 2t, vi) g(t) = (t − 1)u1 (t) − 2(t − 2)u2 (t) + (t − 3)u3 (t)

iii) g(t) = f (t − π)uπ (t); onde f (t) = t2 , v) g(t) = f (t − 1)u2 (t); onde f (t) = 2t,

4.- Use Transformada de