Book edition 9th

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

Pages 616 pages

ISBN 9780321977069

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${e^{3 x}, e^{5 x}, e^{- x}}$ on $(- \infty, \infty)$

Thus, $\{e^{3 x}, e^{5 x}, e^{- x}\}$ are linearly independent on $(- \infty, \infty)$ .

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 $\{e^{3 x}, e^{5 x}, e^{- x}\} .$

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,

$W [e^{3 x}, e^{5 x}, e^{- x}] = |\begin{matrix} e^{3 x} & e^{5 x} & e^{- x} \\ 3 e^{3 x} & 5 e^{5 x} & - e^{- x} \\ 9 e^{3 x} & 25 e^{5 x} & e^{- x} \end{matrix}|$

Solve the above equation,

$W [e^{3 x}, e^{5 x}, e^{- x}] = e^{3 x} \times e^{5 x} \times e^{- x} |\begin{matrix} 1 & 1 & 1 \\ 3 & 5 & - 1 \\ 9 & 25 & 1 \end{matrix}| = e^{7 x} [1 (30) - 1 (12) + 1 (30)] = e^{7 x} [[48]] = 48 e^{7 x}$

Step 2:Check the linearly independent or dependent

Since the above result is $48 e^{7 x} \neq 0$ $(\forall x)$ .

Therefore, $\{e^{3 x}, e^{5 x}, e^{- x}\}$ are linearly independent on $(- \infty, \infty)$ .

Find Study Materials for

Create Study Materials

Select your language

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.
${e^{3 x}, e^{5 x}, e^{- x}}$ on $(- \infty, \infty)$

Step by Step Solution

Step 1:Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

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.{e3x, e5x, e-x} on (-∞, ∞)

Step by Step Solution

Step 1:Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${e^{3 x}, e^{5 x}, e^{- x}}$ on $(- \infty, \infty)$