Basic Laplace Transforms

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The laplace transform has the standard form of:

$F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0}^{\infty }e^{-st}f(t)\,dt$ (Cited From Fullerton, Colby)

However, in this class applying the standard form exclusively to solve problems is not practical. The use of Laplace transform properties greatly simplifies problems. These properties are listed in the book on page 525. The simple properties are listed below and as imported images from mathcad.

Linearity

Example:

If F(s)=(s+2)/(S+1) and f(t)=0 for t<0, then find the Laplace transform for the following functions identifying each property used to compute answers.

(a) $g_{1}(t)=5f(t-2)$ (b)$g_{2}(t)=5(e^{-2t})f(t)$

(a) Linearity: $F(s)=(5(s+2))/(s+1)$

Time Shift: ${\mathcal {L}}\left\{f(t-2)u(t-2)\right\}=e^{-2s}F(s)$ = $(5e^{-2s}(s+2))/(s+1)$

(b) Linearity: $a_{1}F(s)={5(s+2)}/(s+1)$

    Frequency Shift:

${\mathcal {L}}\left\{e^{-at}f(t)\right\}=F(s+a)$

$(5((s+2)+2)))/((s+2)+1)=(5s+20)/(s+1)$