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Q11E

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 326

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.
${x^{- 1}, x^{\frac{1}{2}}, x}$ on $(0, \infty)$

Therefore, $\{x^{- 1}, x^{\frac{1}{2}}, x\}$ are linearly independent on $(0, \infty)$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Using the concept of Wronskian

The given function is $\{x^{- 1}, x^{\frac{1}{2}}, x\} .$

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,

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

Solve the above equation,

$W [x^{- 1}, x^{\frac{1}{2}}, x] = x^{- 1} (0 + \frac{1}{4} x^{- \frac{3}{2}}) - x^{\frac{1}{2}} (0 - 2 x^{- 3}) + x (\frac{1}{4} x^{- 2 - \frac{3}{2}} - x^{- 3 - \frac{1}{2}}) = \frac{1}{4} (x^{- \frac{5}{2}}) + (2 x^{\frac{- 5}{2}}) + (\frac{1}{4} x^{1 - \frac{7}{2}} - x^{1 - \frac{7}{8}}) = \frac{9}{4} (x^{- \frac{5}{2}}) - \frac{3}{4} x^{- \frac{5}{2}} = \frac{6}{4} (x^{- \frac{5}{2}}) = \frac{3}{2} (x^{- \frac{5}{2}})$

Step 2:Check the linearly independent or dependent

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

Therefore, $\{x^{- 1}, x^{\frac{1}{2}}, x\}$ are linearly independent on $(0, \infty)$ .

Most popular questions for Math Textbooks

. find a differential operator that annihilates the given function.

$x^{4} - x^{2} + 11$

find a general solution to the given equation. $y''' + y'' - 5 y' + 3 y = e^{- x} + s i n x .$

use the annihilator method to determinethe form of a particular solution for the given equation.

$y'' + 2 y' + 2 y = e^{- x} c o s x + x^{2}$

find a differential operator that annihilates the given function.

$e^{5 x}$

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

$y''' - \sqrt{x} y = sinx y (π) = 0, y' (π) = 11, y'' (π) = 3$

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