Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t)

Tazmin Horton 2021-09-15 Answered
Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
F(s)=0estf(t)dt
where we assume s is a positive real number. For example, to find the Laplace transform of f(t)=et, the following improper integral is evaluated using integration by parts:
F(s)=0estetdt=0e(s+1)tdt=1s+1
Verify the following Laplace transforms, where u is a real number.
f(t)=cosatF(s)=ss2+a2
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Expert Answer

FieniChoonin
Answered 2021-09-16 Author has 102 answers
Step 1
We have given the function.
f(t)=cosat
Step 2
To prove this we plug the values in the formula.
Lft{cosatright}(s)=0+estcosatdt
=limL0Lestcosatdt
=limL[est(scosat+asinat)(s)2+a2]0L
=limL(esL(scosaL+asinaL)s2+a2es×0(scos(0×a)+asin(0×a))s2+a2)
=limL(scos(0×a)asin(0×a)s2+a2esL(scosaL+asinaL)s2+a2)
=scos(0×a)asin(0×a)s2+a20
=scos0asin0s2+a2
=ss2+a2
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