136Chapter 8The Laplace and Hankel Transforms with ApplicationsSolutions to Exercises 8.11.|11 cos3t|≤11, so (2) holds if you takeM= 11 andaany positive number, saya= 1. Note that (2) also holds witha=0.5.|sinh3t|=|(e3t-e-3t)/2(e3t+e3t)/2=e3t. So (2) holds withM= 1 anda=3.9.Using linearity of the Laplace transform and results from Examples 1 and 2, wehaveL(√t+1√t)(s)=L(t1/2)+L(t-1/2)=Γ(3/2)s3/2+Γ(1/2)s1/2Now Γ(1/2) =√π, so Γ(3/2) = (1/2)Γ(1/2) =√π/2. ThusL(√t+1√t)(s√π2s3/2+±πs.13.Use Example 3 and Theorem 4:L(tsin4t)(s-ddsL(sin(4t)) =-dds4s2+42=8s(s22)217.We haveL(e2tsin 3t)(sL(sin3t)(s-2) =3(s-2)2+323(s-2)2+921.L((t
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