Transformada de Laplace - Exercícios Resolvidos e Aplicações

Determine a transformada de Laplace da função xt e3t cos2t ut a Xs s3 s32 4 b Xs s 4 c Xs s3 s32 7 d Xs 1 s4 e Xs s3 s32 4 Determine a transformada de Laplace da função xt t cosω₀t ut a Xs s2 ω₀2 s2 ω₀2 b Xs s2 ω₀2 s2 ω₀22 c Xs s2 ω₀2 s2 ω₀22 d Xs s2 ω₀2 s2 ω₀22 e Xs s2 ω₀2 s2 ω₀22 Determine a transformada de Laplace da função xt t et ut a Xs 2 s12 b Xs 1 2s12 c Xs 1 s13 d Xs s s12 e Xs 1 s12 Determine a transformada de Laplace da função xt e2t 2etut a Xs 5 s s 2s 2 b Xs s 5 s 1s 2 c Xs 5 s s 1s 2 d Xs 5 s s 1s 2 e Xs s 5 s 1s 1 Determine a transformada de Laplace da função xt ut ut 1 a Xs 1 s21 es b Xs 1 s1 es c Xs 3 s1 es d Xs 1 2s1 es e Xs 1 s1 es 1 Da tabela temos que Leatut 1 s a Logo para a 2 Le2tut 1 s 2 Para a 1 Letut 1 s 1 Utilizando a linearidade temos Xs Le2t 2etut Xs 1 s 2 2 1 s 1 Xs s 1 2 s 2 s 2s 1 Xs s 5 s 2s 1 Gab D 2 Saíemos que Lcos a t s s2 a2 Para a 2 logo Lcos 2 t s s2 4 Além disso temos a seguinte propriedade ect ft Fs c Assim fazendo ft cos 2 t e c 3 temos Le3t cos 2 t ut s 3 s 32 4 Xs s 3 s 32 4 Gabarito A 3 Saíemos que Lcos w0 t ut s s2 w02 Além disso temos que Ltn ft Fns Fazendo n 1 e ft cos w0 t ut temos Lt cosw0 t ut dds s s2 w02 Logo derivando pela regra do quociente Xs 1s2 w02 s2s s2 w02 Xs s2 w02 s2 w02 Gabarito C 4 Temos que Leat 1 s a Logo fazendo a 1 Let ut 1 s 1 Além disso Ltn ft ut fns Logo fazendo n 1 e ft et ut Lt et ut dds 1 s 1 Xs 0 s 1 11 s 12 Xs 1 s 12 Gabarito E 5 Da definição temos que Lut 1s Além disso Lstcutc ecs Fs Fazendo stc 1 e c 1 temos Lut1 es 1s Portanto Xs Lut ut1 Xs 1s ess Xs 1s 1 es Gabarito E