g(t)={0, 0<t<11, 1<t<36, 3<t<54, 5<t g(t)={1,3}(t)+6prod_{3,5}(t)+4u(t-5) compute the Laplace transform g(t)

coexpennan 2021-01-24 Answered

g(t)={00<t<111<t<363<t<545<t 

 g(t)=1,3(t)+63,5(t)+4u(t5)
compute the Laplace transform of g(t)

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Expert Answer

unett
Answered 2021-01-25 Author has 119 answers
Step 1
According to the given information it is required to compute Laplace transform of g(t), where given that g(t) is expressed as,
g(t)=1,3(t)+63,5(t)+4u(t5)
Also from window , for f(t)=a,b(t)
L(f)(s)=easebss,s>0
 and for f(t)=u(ta)
L(f(s))=eass,s>0
Step 2
Laplace transform of g(t) can be expressed as:
g(t)=L(g)(t)
=L[g1(t)+6g2(t)+3g3(t)]
=L(g1)(t)+6L(g2)(t)+4L(g3)(t)
since given L[f+g]=L{f}+L{g}, and L{cf}=cL{f}
Where, g1(t)=1,3(t)
g2(t)=3,5(t)
And g3(t)=u(t5)
Step 3
Now, find
L(g1)(t),L(g2)(t) and L(g3)(t)
As,
L(g1)(t)=e1te3tt,t>0 (comparing by formula mentioned in previous step a=1 , b=3)
L(g2)(t)=e3te5tt,t>0 (similarly here a=3 , b=5)
L(g3)(t)=e5tt,t>0 (And here a=5)
Step 4
Substitute above values in formula for Laplace of g(t).
g(t)=L{(g)(t)}
=L(g1)(t)+6L(g2)(t)+4L(g3)(t)
=ete3tt+6(e3te5tt)+4(e5tt),t>0
=ete3tt+6e3t6e5tt+4e5tt,t>0
=ete3t+6e3t6e5t+4e5tt
=et+5e3t2e5tt,t>0
Thus , Laplace transform of g(t) is
=et+5e3t2e5tt,t>0
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