Laplace transforms A powerful tool in solving problems inengineering and physics is the Laplace transform.

tinfoQ 2020-10-18 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=1(s+1)
Verify the following Laplace transforms, where u is a real number.
f(t)=1F(s)=1s

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Expert Answer

stuth1
Answered 2020-10-19 Author has 97 answers
Step 1
f(t)=1F(s)=1s
Step 2
Given a function f(t), the Laplace transform is a new function F(s) is defined by
F(s)=0estf(t)dt where we assume s is a positive real number.
For f(t)=1, the Laplace transform is given by
F(s)=0est1dt,
Let u=stdu=sdtdt=dus
For t=0,u=s0=0.
For t=,u=s=
Therefore,
F(s)=0(eu)sdu
=0(eu)sdu[abf(x)dx=baf(x)dx]
=1s0(eu)du
=1s[eu]0
=1s[e0e]
=1s[10]
=1s
Hence, verified.
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