DIFFERENTIAL EQUATIONGiven f(t)=-frac{1}{2t}+8 , 0leq t<4 , f(t+4)=f(t)Find F(s)=Lleft{f(t)right} of the Periodic Function

SchachtN 2020-10-26 Answered

DIFFERENTIAL EQUATION
Given f(t)=12t+8,0t<4,f(t+4)=f(t)
Find F(s)=L{f(t)} of the Periodic Function

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Expert Answer

Talisha
Answered 2020-10-27 Author has 93 answers

Step 1
According to the given information it is required to find
F(s)=L{f(t)} Where, L represents Laplace Transformation and given:
f(t)=12t+8,0t<4,f(t+4)=f(t)
Step 2
Laplace transformation of a periodic function with period p>0 is given by
L{f(t)}=L{f1(t)}11esp
L{f(t)}=0pf(t)estdt1esp
here, f1(t) is one period of the function.
Step 3
Calculate the required integral to find the Laplace transformation of given periodic function, as
0pf(t)estdt=04(12t+8)estdt
=1204(t16)estdt
=12[(t16)estdt(ddt(t16)estdt)dt]04
=12[(t16)ests(ests)dt]04=12[t16sest1ests2]04
=12[416se4s+016se0s1e4ss2+1e0ss2]=12[12se4s16s1e4ss2+1s2]
=16s12s2(121)e4ss2
Step 4
Thus,
F(s)=L{f(t)}=L{f1(t)}11esp=0pf(t)estdt1esp
=16s12s2(1e4s)(121)e4ss2(1e4s)
F(s)=16s12s2(1e4s)(121)e4ss2(1e4s)

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