{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ch06

# Differential Equations: An Introduction to Modern Methods and Applications

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 6 Section 6.1 1. (a) A + 3 B = ï£« ï£­ e t 2 e- t e 2 t 2 e t e- t- e 2 t- e t 3 e- t 2 e 2 t ï£¶ ï£¸ + ï£« ï£­ 6 e t 3 e- t 9 e 2 t- 3 e t 6 e- t 3 e 2 t 9 e t- 3 e- t- 3 e 2 t ï£¶ ï£¸ = ï£« ï£­ 7 e t 5 e- t 10 e 2 t- e t 7 e- t 2 e 2 t 8 e t- e 2 t ï£¶ ï£¸ . (b) AB = ï£« ï£­ 2 e 2 t- 2 + 3 e 3 t 1 + 4 e- 2 t- e t 3 e 3 t + 2 e t- e 4 t 4 e 2 t- 1- 3 e 3 t 2 + 2 e- 2 t + e t 6 e 3 t + e t + e 4 t- 2 e 2 t- 3 + 6 e 3 t- 1 + 6 e- 2 t- 2 e t- 3 e 3 t + 3 e t- 2 e 4 t ï£¶ ï£¸ . (c) d A dt = ï£« ï£­ e t- 2 e- t 2 e 2 t 2 e t- e- t- 2 e 2 t- e t- 3 e- t 4 e 2 t ï£¶ ï£¸ . (d) Z A ( t ) dt = ï£« ï£­ e t- 2 e- t e 2 t / 2 2 e t- e- t- e 2 t / 2- e t- 3 e- t e 2 t ï£¶ ï£¸ + C . Therefore, Z 1 A ( t ) dt = ï£« ï£­ e- 2 e- 1 e 2 / 2 2 e- e- 1- e 2 / 2- e- 3 e- 1 e 2 ï£¶ ï£¸- ï£« ï£­ 1- 2 1 / 2 2- 1- 1 / 2- 1- 3 1 ï£¶ ï£¸ = ï£« ï£­ e- 1 2- 2 e- 1 e 2 / 2- 1 / 2 2 e- 2 1- e- 1 1 / 2- e 2 / 2 1- e 3- 3 e- 1 e 2- 1 ï£¶ ï£¸ . 2. First, we note that x = ï£« ï£­- 6 8 4 ï£¶ ï£¸ e- t + ï£« ï£­ 4- 4 ï£¶ ï£¸ e 2 t . At the same time, we calculate ï£« ï£­ 1 1 1 2 1- 1- 1 1 ï£¶ ï£¸ x = ï£« ï£­ 1 1 1 2 1- 1- 1 1 ï£¶ ï£¸ ï£« ï£­ 6- 8- 4 ï£¶ ï£¸ e- t + ï£« ï£­ 1 1 1 2 1- 1- 1 1 ï£¶ ï£¸ ï£« ï£­ 2- 2 ï£¶ ï£¸ e 2 t = ï£« ï£­- 6 8 4 ï£¶ ï£¸ e- t + ï£« ï£­ 4- 4 ï£¶ ï£¸ e 2 t . 3. First, we see that Î¨ = ï£« ï£­ e t- 2 e- 2 t 3 e 3 t- 4 e t 2 e- 2 t 6 e 3 t- e t 2 e- 2 t 3 e 3 t ï£¶ ï£¸ . 1 At the same time, ï£« ï£­ 1- 1 4 3 2- 1 2 1- 1 ï£¶ ï£¸ Î¨ = ï£« ï£­ 1- 1 4 3 2- 1 2 1- 1 ï£¶ ï£¸ ï£« ï£­ e t e- 2 t e 3 t- 4 e t- e- 2 t 2 e 3 t- e t- e- 2 t e 3 t ï£¶ ï£¸ = ï£« ï£­ e t- 2 e- 2 t 3 e 3 t- 4 e t 2 e- 2 t 6 e 3 t- e t 2 e- 2 t 3 e 3 t ï£¶ ï£¸ . 4. Let x 1 = y , x 2 = y , x 3 = y 00 and x 4 = y 000 . Then x 1 = y = x 2 x 2 = y 00 = x 3 x 3 = y 000 = x 4 x 4 = y 0000 =- 4 y 000- 3 y + t =- 4 x 4- 3 x 1 + t. Therefore, ï£« ï£¬ ï£¬ ï£­ x 1 x 2 x 3 x 4 ï£¶ ï£· ï£· ï£¸ = ï£« ï£¬ ï£¬ ï£­ 1 0 0 1 0 0 1- 3 0 0- 4 ï£¶ ï£· ï£· ï£¸ ï£« ï£¬ ï£¬ ï£­ x 1 x 2 x 3 x 4 ï£¶ ï£· ï£· ï£¸ + ï£« ï£¬ ï£¬ ï£­ t ï£¶ ï£· ï£· ï£¸ . 5. Let x 1 = y , x 2 = y and x 3 = y 00 . Then x 1 = y = x 2 x 2 = y 00 = x 3 x 3 = y 000 =- sin t t y 00- 3 t y + cos t =- sin t t x 3- 3 t x 1 + cos t. Therefore, ï£« ï£­ x 1 x 2 x 3 ï£¶ ï£¸ = ï£« ï£­ 1 1- 3 t- sin t t ï£¶ ï£¸ ï£« ï£­ x 1 x 2 x 3 ï£¶ ï£¸ + ï£« ï£­ cos t ï£¶ ï£¸ . 6. Let x 1 = y , x 2 = y , x 3 = y 00 and x 4 = y 000 . Then x 1 = y = x 2 x 2 = y 00 = x 3 x 3 = y 000 = x 4 x 4 = y 0000 =- e t t ( t- 1) y 00- 4 t 2 t ( t- 1) y =- e t t ( t- 1) x 3- 4 t 2 t ( t- 1) x 1 . Therefore, ï£« ï£¬ ï£¬ ï£­ x 1 x 2 x 3 x 4 ï£¶ ï£· ï£· ï£¸ = ï£« ï£¬ ï£¬ ï£­ 1 1 1- 4 t 2 t ( t- 1)- e t t ( t- 1) ï£¶ ï£· ï£· ï£¸ ï£« ï£¬ ï£¬ ï£­ x 1 x 2 x 3 x 4 ï£¶ ï£· ï£· ï£¸ . 7. Let x 1 = y , x 2 = y , and x 3 = y 00 . Then x 1 = y = x 2 x 2 = y 00 = x 3 x 3 = y 000 =- ty 00- t 2 y- t 2 y + ln t =- tx 3- t 2 x 2- t 2 x 1 + ln t....
View Full Document

{[ snackBarMessage ]}

### Page1 / 101

ch06 - Chapter 6 Section 6.1 1(a A 3 B = ï£ ï£­ e t 2 e t e...

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

View Full Document
Ask a homework question - tutors are online