g(t)={00&lt;t&lt;111&lt;t&lt;363&lt;t&lt;545&lt;t g(t)=∏1,3(t)+6∏3,5(t)+4u(t−5) compute the Laplace transform of g(t)

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g(t)={00&amp;lt;t&amp;lt;111&amp;lt;t&amp;lt;363&amp;lt;t&amp;lt;545&amp;lt;t    g(t)=∏1,3(t)+6∏3,5(t)+4u(t−5) compute the Laplace transform of g(t)

unett · Accepted Answer

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)+6∏3,5(t)+4u(t−5)  Also from window , for f(t)=∏a,b(t)  L(f)(s)=e−as−e−bss,s&amp;gt;0   and for f(t)=u(t−a)  L(f(s))=e−ass,s&amp;gt;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(t−5)  Step 3  Now, find  L(g1)(t),L(g2)(t) and L(g3)(t)  As,   L(g1)(t)=e−1t−e−3tt,t&amp;gt;0   (comparing by formula mentioned in previous step a=1 , b=3)  L(g2)(t)=e−3t−e−5tt,t&amp;gt;0 (similarly here a=3 , b=5)  L(g3)(t)=e−5tt,t&amp;gt;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)  =e−t−e−3tt+6(e−3t−e−5tt)+4(e−5tt),t&amp;gt;0  =e−t−e−3tt+6e−3t−6e−5tt+4e−5tt,t&amp;gt;0  =e−t−e−3t+6e−3t−6e−5t+4e−5tt  =e−t+5e−3t−2e−5tt,t&amp;gt;0  Thus , Laplace transform of g(t) is  =e−t+5e−3t−2e−5tt,t&amp;gt;0

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)

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