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Lecture Notes in Differential Equations - Bruce E. Shapiro

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347<br />

Translations <strong>in</strong> the Laplace Variable<br />

If F (s) is the Laplace Transform of f(t),<br />

F (s) =<br />

∫ ∞<br />

0<br />

f(t)e −st dt (32.128)<br />

What happens when we shift the s-variable a distance a?<br />

S = s − a,<br />

Substitut<strong>in</strong>g<br />

F (s − a) = F (S) =<br />

=<br />

=<br />

∫ ∞<br />

0<br />

∫ ∞<br />

0<br />

∫ ∞<br />

which is often more useful <strong>in</strong> its <strong>in</strong>verse form,<br />

0<br />

f(t)e −St dt (32.129)<br />

f(t)e −(s−a)t dt (32.130)<br />

e at f(t)e −st dt (32.131)<br />

= L [ e at f(t) ] (32.132)<br />

L −1 [F (s − a)] = e at f(t) (32.133)<br />

Example 32.13. F<strong>in</strong>d f(t) such that F (s) = s + 2<br />

(s − 2) 2 .<br />

Us<strong>in</strong>g partial fractions,<br />

Hence<br />

s + 2<br />

(s − 2) 2 = A<br />

s − 2 + B<br />

(s − 2) 2 (32.134)<br />

A(s − 2)<br />

=<br />

(s − 2) 2 + B<br />

(s − 2) 2 (32.135)<br />

s + 2 = As + (B − 2A) (32.136)<br />

A = 1 (32.137)<br />

B − 2A = 2 =⇒ B = 4 (32.138)<br />

Therefore<br />

F (s) = s + 2<br />

(s − 2) 2 = 1<br />

s − 2 + 4<br />

(s − 2) 2 (32.139)

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