{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

nagle_differential_equations_ISM_Part42

# nagle_differential_equations_ISM_Part42 - Exercises 7.3...

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

Exercises 7.3 EXERCISES 7.3: Properties of the Laplace Transform 2. Using the linearity of the Laplace transform, we get L 3 t 2 - e 2 t ( s ) = 3 L t 2 ( s ) - L e 2 t ( s ) . From Table 7.1 in Section 7.2 we know that L t 2 ( s ) = 2! s 3 = 2 s 3 , L e 2 t ( s ) = 1 s - 2 . Thus L 3 t 2 - e 2 t ( s ) = 3 2 s 3 - 1 s - 2 = 6 s 3 - 1 s - 2 . 4. By the linearity of the Laplace transform, L 3 t 4 - 2 t 2 + 1 ( s ) = 3 L t 4 ( s ) - 2 L t 2 ( s ) + L { 1 } ( s ) . From Table 7.1 of the text we see that L t 4 ( s ) = 4! s 5 , L t 2 ( s ) = 2! s 3 , L { 1 } ( s ) = 1 s , s > 0 . Therefore, L 3 t 4 - 2 t 2 + 1 ( s ) = 3 4! s 5 - 2 2! s 3 + 1 s = 72 s 5 - 4 s 3 + 1 s , is valid for s > 0. 6. We use the linearity of the Laplace transform and Table 7.1 to get L e - 2 t sin 2 t + e 3 t t 2 ( s ) = L e - 2 t sin 2 t ( s ) + L e 3 t t 2 ( s ) = 2 ( s + 2) 2 + 4 + 2 ( s - 3) 3 , s > 3 . 8. Since (1 + e - t ) 2 = 1 + 2 e - t + e - 2 t , we have from the linearity of the Laplace transform that L (1 + e - t ) 2 ( s ) = L { 1 } ( s ) + 2 L e - t ( s ) + L e - 2 t ( s ) . From Table 7.1 of the text, we get L { 1 } ( s ) = 1 s , L e - t ( s ) = 1 s + 1 , L e - 2 t ( s ) = 1 s + 2 . Thus L (1 + e - t ) 2 ( s ) = 1 s + 2 s + 1 + 1 s + 2 , s > 0 . 201

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

View Full Document
Chapter 7 10. Since L e 2 t cos 5 t ( s ) = s - 2 ( s - 2) 2 + 25 , we use Theorem 6 to get L te 2 t cos 5 t ( s ) = L t ( e 2 t cos 5 t ) ( s ) = - L e 2 t cos 5 t ( s ) = - s - 2 ( s - 2) 2 + 25 = - [( s - 2) 2 + 25] - ( s - 2) · 2( s - 2) [( s - 2) 2 + 25] 2 = ( s - 2) 2 - 25 [( s - 2) 2 + 25] 2 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

nagle_differential_equations_ISM_Part42 - Exercises 7.3...

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

View Full Document
Ask a homework question - tutors are online