Q12E Determine whether the given func... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 6: Q12E (page 326)

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

Short Answer

Expert verified

Therefore, the function $\{\cos 2 x, 鈥� \cos^{2} x, 鈥� \sin^{2} x\}$ is linearly dependent on $(- 鈭�, 鈥� 鈭�)$ .

Step by step solution

Using the concept of Wronskian

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

Apply the concept of Wronskian,

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

Therefore,

$W [\cos 2 x, \cos^{2} x, \sin^{2} x] = W [\cos 2 x, \frac{1 + \cos 2 x}{2}, \frac{1 - \cos 2 x}{2}] = |\begin{matrix} \cos 2 x & \frac{1 + \cos 2 x}{2} & \frac{1 - \cos 2 x}{2} \\ - 2 \sin 2 x & - \sin 2 x & \sin 2 x \\ - 4 \cos 2 x & - 2 \cos 2 x & 2 \cos 2 x \end{matrix}|$

Solve the above equation,

$W [\cos 2 x, 鈥� \cos^{2} x, 鈥� \sin^{2} x] = \cos 2 x (- 2 \sin 2 xcos 2 x + 2 \sin 2 xcos 2 x) - \frac{1 + \cos 2 x}{2} (- 4 \sin 2 xcos 2 x + 4 \sin 2 xcos 2 x) + \frac{1 - \cos 2 x}{2} (4 \sin 2 xcos 2 x - 4 \sin 2 xcos 2 x) = \cos 2 x (0) - \frac{1 + \cos 2 x}{2} (0) + \frac{1 - \cos 2 x}{2} (0) = 0$

Step 2:Check the linearly independent or dependent

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

Therefore, $\{\cos 2 x, 鈥� \cos^{2} x, 鈥� \sin^{2} x\}$ is linearly dependent on $(- 鈭�, 鈥� 鈭�)$ .

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91影视

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

Short Answer

Step by step solution

Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

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91影视

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

Short Answer

Step by step solution

Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Pure Maths

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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