nagle_differential_equations_ISM_Part47

nagle_differential_equations_ISM_Part47 - Chapter 7 16. We...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 7 16. We apply formula (6) (Theorem 8) with F ( s ) = 1 / ( s 2 + 4) and a = 1. L - 1 ± e - s s 2 + 4 ² ( t ) = L - 1 ± 1 s 2 + 4 ² ( t - 1) u ( t - 1) = sin(2 t - 2) 2 u ( t - 1) .. 18. By partial fractions decomposition, 3 s 2 - s + 2 ( s - 1) ( s 2 + 1) = - 2 s - 1 + s s 2 + 1 so that L - 1 ± e - s (3 s 2 - s + 2) ( s - 1) ( s 2 + 1) ² ( t ) = L - 1 ± 2 e - s s - 1 ² ( t ) + L - 1 ± e - s s s 2 + 1 ² ( t ) = ³ 2 L - 1 ± 1 s - 1 ² ( t - 1) + L - 1 ± s s 2 + 1 ² ( t - 1) ´ u ( t - 1) = µ 2 e t - 1 + cos( t - 1) u ( t - 1) . 20. In this problem, we apply methods of Section 7.5 of solving initial value problems using the Laplace transform. Taking the Laplace transform of both sides of the given equation and using the linear property of the Laplace transform, we get L{ I 00 + 4 I } ( s ) = L{ I 00 } ( s ) + 4 L{ I } ( s ) = L{ g ( t ) } ( s ) . (7.14) Let us denote I ( s ) := L{ I } ( s ). By Theorem 5, Section 7.3, L{ I 00 } ( s ) = s 2 I ( s ) - sI (0) - I 0 (0) = s 2 I ( s ) - s - 3 . Thus, L{ I 00 + 4 I } ( s ) = ( s 2 I ( s ) - s - 3 ) + 4 I ( s ) = ( s 2 + 4 ) I ( s ) - ( s + 3) . (7.15) To find the Laplace transform of g ( t ), we express this function using the unit step function u ( t ). Since g ( t ) identically equals to 3 sin t for 0 < t < 2 π and jumps to 0 at t = 2 π , we can write g ( t ) = (3 sin t ) [1 - u ( t - 2 π )] = 3 [sin t - (sin t ) u ( t - 2 π )] . Therefore, L{ g ( t ) } ( s ) = 3 ³ 1 s 2 + 1 - e - 2 πs s 2 + 1 ´ = 3 (1 - e - 2 πs ) s 2 + 1 226
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Exercises 7.6 Substituting this equation and (7.15) into (7.14) and solving for I ( s ) yields I ( s ) = s s 2 + 4
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.

Page1 / 5

nagle_differential_equations_ISM_Part47 - Chapter 7 16. We...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online