Ejercicios De Transformadas De Laplace

Ejercicios: aplicando definición 1.1. Determine L {f(t)}
1) L et+7=0∞e-st∙et+7dt=0∞e-ts-1∙e7dt=e70∞e-ts-1dt=e7s-1

2) L te4t=0∞e-st∙t∙e4tdt=0∞t∙e-ts-4dt=1s-42

3) Le-tsent=0∞e-st∙e-tsentdt=0∞e-ts+1sentdt=-1s+1e-t(s+1)sent0∞+
+1s+10∞e-ts+1∙costdt=-1s+12e-t(s+1)cost0∞-1s+120∞e-ts+1∙sendt
1+1s+120∞e-ts+1∙sentdt=1s+120∞e-ts+1∙sendt=1s+12+14) L tcos=0∞e-st∙tcostdt=-1se-st∙tcost0∞-1s0∞e-sttsentdt+1s0∞e-stcostdt
=1s2e-st∙tsent0∞-1s20∞e-sttcostdt-1s20∞e-stsentdt+1s0∞e-stcostdt=-1s20∞e-sttcostdt+1s3e-st∙sent0∞-1s30∞e-stcostdt+1s0∞e-stcostdt
0∞e-st∙tcostdt=-1s20∞e-sttcostdt-1s30∞e-stcostdt+1s0∞e-stcostdt
s2+1s20∞e-st∙tcostdt=s2-1s30∞e-stcostdt
0∞e-st∙tcostdt=s2-1s3∙s2s2+1∙ss2+1=s2-1s2+12

5) Lf(t)=-1 0≤t<11 t≥π=0πe-st-1dt+0πe-st1dt=1se-st01-1se-st1∞
L f(t)=1se-s-1-1s0-e-s=1se-s-1s+1se-s=2se-s-1s=2e-s-1s

6) L f(t)=t 0≤t<11t≥1=01te-st+1∞e-st1dt
=-1ste-st01+1s01e-stdt-1se-st1∞=-1se-s-1s2e-st01+1se-s=-1s2e-s+1s2=1-e-ss2

7) L f(t)=sent 0≤t<π0t≥π=0πe-stsentdt=-1se-st∙sent0π+1s0πe-stcostdt
0πe-stsentdt=-1s2e-st∙cost0π-1s20πe-stsentdt=-1s2-e-sπ-1-1s20πe-stsentdt
1+1s20πe-stsentdt=-1s2-e-sπ-10πe-stsentdt=-1s2-e-sπ-1∙s2s2+1=e-sπ+1s2+1

Determine F(S)f(t) según se indique

1) L te10t=-1ddsL e10t=-1dds1s-10=1s-102

2) L t3e-2t=-13d3ds3L e-2t=-d3ds31s-2=d2ds21s-22=-dds2s-23=6s-24

3)L etsent=3s2+9s→s-1=3s-12+9

4) L e5tsenh3t=3s2-9s→s-5=3s-52+9

5) L tet+e2t2=L te2t+2e3t+e4t=L te2t+2te3t+te4t
=L te2t+2te3t+te4t=L te2t+L 2te3t+L te4t==-1ddsL e2t+2-1ddsL e3t+-1ddsL e4t=1s-22+2s-32+1s-42

6) L e-tsen2t=2ss2+4s→s+1=2s+1s+12+4

7) L -11s+23=e-2tL -11s3=12e-2tL -12s3=12e-2tt2=e-2tt22