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 interval $(0, \infty)$ .
(a) ${e^{2 x}, x^{2} e^{2 x}, e^{- x}}$
(b) ${e^{x} s i n 2 x, x e^{x} s i n 2 x, e^{x}, x e^{x}}$
(c) ${2 e^{2 x} - e^{x}, e^{2 x} + 1, e^{2 x} - 3, e^{x} + 1}$

Given set of functions are Linearly Independent (LI).
Given set of functions are Linearly Independent (LI).
Given set of functions are Linearly Independent (LI).

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the given functions are linearly dependent or linearly independent

$< e^{2 x}, x^{2} e^{2 x}, e^{- x} >$

Assuming scalars $c_{1}, c_{2}, c_{3}$ such that

$c_{1} e^{2 x} + c_{2} x^{2} e^{2 x} + c_{3} e^{- x} = 0$

$x = 0 c_{4} = 0 = c_{2} c_{1} + c_{3} = 0$

At $x \to \infty$

$c_{1} (\infty) + c_{2} (\infty) = 0 c_{1} = c_{2} = 0 = c_{3}$

Hence

Given set of functions are Linearly Independent (LI).

Step 2: Determine the given functions are linearly dependent or linearly independent

$< e^{x} \sin 2 x, x e^{x} \sin 2 x, e^{x}, x e^{x} >$

Assuming scalars $c_{1}, c_{2}, c_{3}, c_{4}$ such that

$c_{1} e^{x} \sin 2 x + c_{2} x e^{x} \sin 2 x + c_{3} e^{x} + c_{4} x e^{x} = 0$

At $x \to 0$

$c_{4} = 0 = c_{2}$

At x = 0

$c_{3} = 0 c_{1} = c_{2} = c_{3} = c_{4} = 0$

Hence

Given set of functions are Linearly Independent (LI).

Step 3: Determine the given functions are linearly dependent or linearly independent

$< 2 e^{2 x} - e^{x}, e^{2 x} + 1, e^{2 x} - 3, e^{x} + 1 >$

Assuming scalars $c_{1}, c_{2}, c_{3}, c_{4}$ such that

$c_{1} (2 e^{2 x} - e^{x}) + c_{2} (e^{2 x} + 1) + c_{3} (e^{2 x} - 3) + c_{4} (e^{x} + 1) = 0$

At $x \to - \infty$

$c_{2} - 2 c_{3} + c_{4} = 0$

At x = 0

$c_{1} + 2 c_{2} - 2 c_{3} + 2 c_{4} = 0$

At x = 1

$e^{2} (2 c_{1} + c_{2} + c_{3}) = c_{1} - c_{1}$

At x = -1

$2 c_{1} + c_{2} + c_{3} + c_{(c_{1}} - c_{1}) = 0$

From equation third and fourth we get

$c_{2} = - c_{1} c_{4} = c_{1} c_{3} = - c_{1}$

From equation second, we can get

$c_{1} = 0 c_{1} = c_{2} = c_{3} = c_{4} = 0$

Hence

Given set of functions are Linearly Independent (LI).

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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 interval $(0, \infty)$ .
(a) ${e^{2 x}, x^{2} e^{2 x}, e^{- x}}$
(b) ${e^{x} s i n 2 x, x e^{x} s i n 2 x, e^{x}, x e^{x}}$
(c) ${2 e^{2 x} - e^{x}, e^{2 x} + 1, e^{2 x} - 3, e^{x} + 1}$

Step by Step Solution

Step 1: Determine the given functions are linearly dependent or linearly independent

Step 2: Determine the given functions are linearly dependent or linearly independent

Step 3: Determine the given functions are linearly dependent or linearly independent

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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 interval (0,∞) .(a) {e2x,x2e2x,e-x}(b) {exsin2x,xexsin2x,ex,xex}(c) {2e2x-ex,e2x+1,e2x-3,ex+1}

Step by Step Solution

Step 1: Determine the given functions are linearly dependent or linearly independent

Step 2: Determine the given functions are linearly dependent or linearly independent

Step 3: Determine the given functions are linearly dependent or linearly independent

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Pure Maths

Determine whether the given functions are linearly dependent or linearly independent on the interval $(0, \infty)$ .
(a) ${e^{2 x}, x^{2} e^{2 x}, e^{- x}}$
(b) ${e^{x} s i n 2 x, x e^{x} s i n 2 x, e^{x}, x e^{x}}$
(c) ${2 e^{2 x} - e^{x}, e^{2 x} + 1, e^{2 x} - 3, e^{x} + 1}$