Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${\sin^{2} x, \cos^{2} x, 1}$ on $(- \infty, \infty)$

Thus, the function is $\{\sin^{2} x, \cos^{2} x, 1\}$ linearly dependent on $(- \infty, \infty)$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Using the concept of Wronskian

The given function is $\{\sin^{2} x, \cos^{2} x, 1\} .$

Apply the concept of Wronskian,

$W [f_{1}, f_{2}, \dots, f_{n}] = |\begin{matrix} f_{1} (x) & f_{2} (x) & \dots & f_{n} (x) \\ f_{1}' (x) & f_{2}' (x) & \dots & f_{n}' (x) \\ ⋮ & ⋮ & ⋮ \\ {f_{1}}^{n - 1} (x) & {f_{2}}^{n - 1} (x) & \dots & {f_{n}}^{n - 1} (x) \end{matrix}|$

Therefore,

$W [\sin^{2} x, \cos^{2} x, 1] = W [\frac{1 - \cos 2 x}{2}, \frac{1 + \cos 2 x}{2}, 1] = |\begin{matrix} \frac{1 - \cos 2 x}{2} & \frac{1 + \cos 2 x}{2} & 1 \\ \sin 2 x & - \sin 2 x & 0 \\ 2 \cos 2 x & - 2 \cos 2 x & 0 \end{matrix}|$

Solve the above equation,

$W [\sin^{2} x, \cos^{2} x, 1] = \frac{1 - \cos 2 x}{2} (0) - \frac{1 + \cos 2 x}{2} (0) + 1 (- 2 \sin 2 xcos 2 x + 2 \sin 2 xcos 2 x) = 0$

Step 2:Check the linearly independent or dependent

The above function is equal to zero $(\forall x) .$

Therefore, $\{\sin^{2} x, \cos^{2} x, 1\}$ are linearly dependent on $(- \infty, \infty)$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${\sin^{2} x, \cos^{2} x, 1}$ on $(- \infty, \infty)$

Step by Step Solution

Step 1:Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.{sin2x, cos2x, 1} on (-∞, ∞)

Step by Step Solution

Step 1:Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${\sin^{2} x, \cos^{2} x, 1}$ on $(- \infty, \infty)$