hw5solns - Section 3.2 3. { e 2 x , xe 2 x } . By...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Section 3.2 3. { e 2 x , xe 2 x } . By definition, c 1 e 2 x + c 2 xe 2 x = 0 e 2 x ( c 1 + c 2 x ) = 0 c 1 + c 2 x = 0 c 1 = c 2 = 0 linearly independent Using the Wronskian, det e 2 x xe 2 x 2 e 2 x e 2 x (1 + x ) x =0 = det 1 0 2 1 = 1 linearly independent 7. Using the Wronskian, W ( x ) = det sin 2 x cos 2 x 2 cos 2 x- 2 sin 2 x W (0) = det 0 1 2 0 =- 2 6 = 0 . Therefore { sin 2 x, cos 2 x } is linearly independent. 8. c 1 ( x 3- 4) + c 2 x + c 3 3 x = 0. Grouping like terms and choosing c 1 = 0, c 2 =- 3, and c 3 = 1 shows the set is linearly dependent. 22. (a) On [0 , 1], | x | = x . Then c 1 x + c 2 x = 0 when c 1 =- c 2 shows the set is linearly dependent. (b) On [- 1 , 0], | x | =- x . Then c 1 x + c 2 (- x ) when c 1 = c 2 shows the set is linearly dependent. (c) On [- 1 , 1], we need c 1 x + c 2 | x | = 0. This can only happen when c 1 =- c 2 and when c 1 = c 2 . These are both true only if c 1 = c 2 = 0. Thus the set is linearly independent on [- 1 ,...
View Full Document

Page1 / 2

hw5solns - Section 3.2 3. { e 2 x , xe 2 x } . By...

This preview shows document pages 1 - 2. Sign up to view the full document.

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