Chem Differential Eq HW Solutions Fall 2011 136

Chem Differential Eq HW Solutions Fall 2011 136 - 136...

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136 Chapter 8 The Laplace and Hankel Transforms with Applications Solutions to Exercises 8.1 1. | 11 cos3 t |≤ 11, so (2) holds if you take M = 11 and a any positive number, say a = 1. Note that (2) also holds with a =0 . 5. | sinh3 t | = | ( e 3 t - e - 3 t ) / 2 ( e 3 t + e 3 t ) / 2= e 3 t . So (2) holds with M = 1 and a =3 . 9. Using linearity of the Laplace transform and results from Examples 1 and 2, we have L ( t + 1 t )( s )= L ( t 1 / 2 )+ L ( t - 1 / 2 ) = Γ(3 / 2) s 3 / 2 + Γ(1 / 2) s 1 / 2 Now Γ(1 / 2) = π , so Γ(3 / 2) = (1 / 2)Γ(1 / 2) = π/ 2. Thus L ( t + 1 t )( s π 2 s 3 / 2 + ± π s . 13. Use Example 3 and Theorem 4: L ( t sin4 t )( s - d ds L (sin(4 t )) = - d ds 4 s 2 +4 2 = 8 s ( s 2 2 ) 2 17. We have L ( e 2 t sin 3 t )( s L (sin3 t )( s - 2) = 3 ( s - 2) 2 +3 2 3 ( s - 2) 2 +9 21. L (( t
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