Answer to Question #177423 in Differential Equations for Promise Omiponle

Question #177423

Use Definition 7.1.1.

DEFINITION 7.1.1    Laplace Transform

Let f be a function defined for t ≥ 0.

 Then the integral

{f(t)} = ∞

estf(tdt0

is said to be the Laplace transform of f, provided that the integral converges.

Find {f(t)}.

 (Write your answer as a function of s.)

f(t) = {t, 0 ≤ t < 1

{1,  t ≥ 1


1
Expert's answer
2021-04-15T06:56:27-0400

L(f(t))=0estf(t)dt=ℒ(f(t))=\int_0^{\infty}e^{-st}f(t) dt=


=01esttdt+1estdt=\int_0^1e^{-st}t dt+\int_1^{\infty}e^{-st}dt


01esttdt=1sestt01+01estsdt=\int_0^1e^{-st}tdt=-\frac{1}{s}e^{-st}t|_0^1+\int_0^1\frac{e^{-st}}{s}dt=


=1ses1s2est01==-\frac{1}{s}e^{-s}-\frac{1}{s^2}e^{-st}|_0^1=


=1ses1s2(es1)=-\frac{1}{s}e^{-s}-\frac{1}{s^2}(e^{-s}-1)


1estdt=1sest1=1ses\int_1^{\infty}e^{-st}dt=-\frac{1}{s}e^{-st}|_1^{\infty}=\frac{1}{s}e^{-s}


L(f(t))=1ses1s2(es1)+1ses=ℒ(f(t))=-\frac{1}{s}e^{-s}-\frac{1}{s^2}(e^{-s}-1)+\frac{1}{s}e^{-s}=


=1s2(1es)=\frac{1}{s^2}(1-e^{-s})


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