Lecture Notes in Differential Equations - Bruce E. Shapiro

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Example 32.18. Find the Laplace transform of the full-wave rectification
of sin t,
{
sin t 0 ≤ t < π
f(t) =
(32.202)
f(t − π) t ≥ π
Using the formula for the transform of a periodic function,
∫
1 π
F (s) =
1 − e −πs sin te −st dt (32.203)
0
π
1 1
=
1 − e −πs 1 + s 2 e−st (−s sin t − cos t)
∣ (32.204)
using (A.105) to find the integral. Hence
1 1
F (s) =
1 − e −πs 1 + s 2 [e−πs (−s sin π − cos π)
− e 0 (−s sin 0 − cos 0)] (32.205)
1 1
=
1 − e −πs 1 + s 2 [e−πs (1) + (1)] (32.206)
= 1 + e−πs
1 − e −πs · 1
1 + s 2 (32.207)
Example 32.19. Solve y ′′ + 9y = cos 3t, y(0) = 2, y ′ (0) = 5.
The Laplace Transform is
0
s 2 Y (s) − sy(0) − y ′ (0) + 9Y (s) =
s
s 2 + 9
(32.208)
Substituting the initial conditions and grouping,
(9 + s 2 )Y (s) − 2s − 5 = s
s 2 + 9
(32.209)
rearranging terms and solving for Y (s),
(9 + s 2 )Y (s) = 2s + 5 + s
s 2 + 9
(32.210)
Y (s) = 2s + 5
9 + s 2 + s
(9 + s 2 ) 2 (32.211)
= 2 ·
s
9 + s 2 + 5 · 1
9 + s 2 + s
(9 + s 2 ) 2 (32.212)