Q13E Determine whether the given func... [FREE SOLUTION]

91影视

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: Q13E (page 326)

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}}$ on $(- 鈭�, 鈥� 鈭�)$

Short Answer

Expert verified

Linearly independent

Step by step solution

Step 1:Using the concept of Wronskian

The given function is $\{x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}\} .$

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,

$W [x, x^{2}, x^{3}, x^{4}] = |\begin{matrix} x & x^{2} & x^{3} & x^{4} \\ 1 & 2 x & 3 x^{2} & 4 x^{3} \\ 0 & 2 & 6 x & 12 x^{2} \\ 0 & 0 & 6 & 24 x \end{matrix}|$

Solve the above equation,

$W [x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}] = x |\begin{matrix} 2 x & 3 x^{2} & 4 x^{3} \\ 1 & 6 x & 12 x^{2} \\ 0 & 6 & 24 x \end{matrix}| - x^{2} |\begin{matrix} 1 & 3 x^{2} & 4 x^{3} \\ 0 & 6 x & 12 x^{2} \\ 0 & 6 & 24 x \end{matrix}| + x^{3} |\begin{matrix} 1 & 2 x & 4 x^{3} \\ 0 & 1 & 12 x^{2} \\ 0 & 0 & 24 x \end{matrix}| - x^{4} |\begin{matrix} 1 & 2 x & 3 x^{2} \\ 0 & 1 & 6 x \\ 0 & 0 & 6 \end{matrix}| = x (96 x^{3}) - x^{2} (72 x^{2}) + x^{3} (24 x) - x^{4} (6) = 96 x^{4} - 72 x^{4} + 24 x^{4} - 6 x^{4} = 42 x^{4}$

Step 2:Check the linearly independent or dependent

The above function is not equal to zero $(鈭赌 x)$

Therefore, $\{x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}\}$ are linearly independent 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.
${x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}}$ on $(- 鈭�, 鈥� 鈭�)$

Short Answer

Step by step solution

Step 1: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.{x,鈥�x2,鈥�x3,鈥�x4}on(-鈭�,鈥�鈭�)

Short Answer

Step by step solution

Step 1: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

Discrete Mathematics

Applied Mathematics

Pure Maths

Geometry

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x, 鈥� x^{2}, 鈥� x^{3}, 鈥� x^{4}}$ on $(- 鈭�, 鈥� 鈭�)$