Find the Laplace transforms of the following time functions. Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For

Kye 2021-02-21 Answered
Find the Laplace transforms of the following time functions.
Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.
a)f(t)=1+2t b)f(t)=sinωtHint: Use Euler’s relationship, sinωt=e(jωt)e(jωt)2j
c)f(t)=sin(2t)+2cos(2t)+etsin(2t)
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Expert Answer

jlo2niT
Answered 2021-02-22 Author has 96 answers
Step 1 To find the Laplace transform of the following functions.
Laplace transform of a function f(t) can be defined as L(f(t))=0f(t)estdt
Step 2
a) To find the Laplace transform of f(t)=1+2t
L(f(t))=L(1+2t)
=0(1+2t)estdt
=0estdt+0(2t)estdt
=[ests]0+2[(t)ests][1ests2]0
=[01s]+2[(00)(01s2]
=1s+2s2
=s+2s2 Thus, L(f(t))=s+2s2
Step 3
b) To find the Laplace transform of f(t)=sinωt=ejωtejωt2j
L(f(t))=L(sinωt)
=0sinωtestdt
=0ejωtejωt2jestdt =12j0e(jωs)te(jω+s)tdt
=12j[(e(jωs)t)jωs)(e(jω+s)t(jω+s))]0
=12j[(e(sjω)t(sjω))(e(jω+s)t(jω+s))]0
=12j[01(sjω)01(jω+s)]
=12j[1(sjω)1(jω+s)]
=12j[1(sjω)1(s+jω)] =12j[(s+jω)(sjω)(sjω)(s+jω)]
=ωs2(jω)2
=ωs2+ω2 Since ω2=1
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