Transformada de Laplace - Teoria e Exercícios Resolvidos

a Lsen at 0 senat est dt dv u u est dv senat v cosatb du sest Lsen at lim A est cosata A0 λa 0 A est cosat dt u dv u est dv cosat du sest v senata Lsen at 1a λa lim Aest senataA0 λa 0 A est sen at dt Lsen at 1a λ2a2 Lsen at Lsen at 1 λ2a2 1a Lsen at a2 λ2a2 1a Lsen at aa2 λ2 b Lcos at 0 est cos at dt u dv u est dv cosat dt du sest v senata Lcos at lim A est senata 0A λa 0A est senat dt Lcos at λa Lsen at Lcos at λa aa2 λ2 Lcos at λa2 λ2 22 03 23 TRANSFORMADA DE LA PLACE Exercicios Encontre a T L dos funções usando a definição a ft sen at b ft cos at c ft tm d ft sen ln at e ft cos ln at ① Exemplo ft ln t não é Continua na origem nem limitada quando t 02 Entretanto Llnt 58lns s colocar em Lft c Ltn ₀ tn est dt u tn du n tn1 dv est dt v ests Ltn tn ests ₀ ns ₀ tn1 est dt Ltn ns Ltn1 sabemos que Lt⁰1 ₀ est dt ests ₀ 1s Ltn ns Ltn1 Ltn nn1s² Ltn2 Ltn nn1nk1sk Ltnk se k n temos Ltn nsn Lt⁰1 nsn1 d Lsinhat Leat eat2 Lsinhat ₀ eat eat2 est dt Lsinhat 12 ₀ eat est dt ₀ eat est dt Lsinhat 12 ₀ esat dt ₀ east dt Lsinhat 12 esatsa 0 eastsa 0 Lsinhat 12 1sa 1sa 12 2as² a² Lsinhat as² a² e Lcoshat Leat eat2 Lcoshat ₀ eat eat2 est dt Lcoshat 12 ₀ eat est dt ₀ eat est dt Lcoshat 12 ₀ esat dt ₀ east dt Lcoshat 12 esatas 0 eastas 0 Lcoshat 12 1sa 1sa 1s² a² s Lcoshat ss² a² Exemplo Lln t ₀ lnt eSt dt ₀ ln t et dt r x St t xs dt dxs Lln t 1s ₀ lnx lns ex dx Lln t 1s ₀ lnx ex dx lns ₀ ex dx Lln t 1s r lns ex0 Lln t 1s r lns Lln t r lnss