Explanation Given: The given vectors is: [ t 2 2 t ] , [ t | t | 2 | t | ] Procedure used: Wronskian of n vector functions: W [ X 1 , ⋯ , X n ] ( t ) = | x 1 , 1 ( t ) x 1 , 2 ( t ) ⋯ x 1 , n ( t ) x 2 , 1 ( t ) x 2 , 2 ( t ) ⋯ x 2 , n ( t ) ⋮ ⋮ ⋮ x n , 1 ( t ) x n , 2 ( t ) ⋯ x n , n ( t ) | The n vector functions are linearly independent on an interval if determinant is non zero and if determinant are zero, then the vector functions are linearly dependent. Calculation: Consider the determinant: | t 2 t | t | 2 t 2 | t | | ≡ 0 Find the determinant on the interval ( − ∞ , 0 ) : D ( t ) = | t 2 − t 2 2 t − 2 t | = ( t 2 ) ⋅ ( − 2 t ) − ( − t 2 ) ⋅ ( 2 t ) = − 2 t 3 + 2 t 3 = 0 Find the determinant on the interval ( 0 , ∞ ) : D ( t ) = | t 2 t 2 2 t 2 t | = ( t 2 ) ⋅ ( 2 t ) − ( t 2 ) ⋅ ( 2 t ) = 2 t 3 − 2 t 3 = 0 Hence, proved | t 2 t | t | 2 t 2 | t | | ≡ 0 on the interval ( − ∞ , ∞ ) . Consider the equation: c 1 [ t 2 2 t ] + c 2 [ t | t | 2 | t | ] = 0 Consider the interval ( − ∞ , 0 ) : c 1 [ t 2 2 t ] + c 2 [ − t 2 − 2 t ] = 0 Simplify the equation:

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

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

Publisher: PEARSON

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Question

Chapter 9.4, Problem 31E

To determine

To prove:

The determinant |t2t|t|2t2|t||≡0 and the given vectors are linearly independent on the interval (−∞,∞).

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Chapter 9.4, Problem 30E

Chapter 9.4, Problem 32E

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Chapter 9 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

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To prove: The determinant | t 2 t | t | 2 t 2 | t | | ≡ 0 and the given vectors are linearly independent on the interval ( − ∞ , ∞ ) .

Solution Summary: The author explains that the determinant left|cct2 and the given vectors are linearly independent on an interval.

Concept explainers

To prove:

The determinant |t2t|t|2t2|t||≡0 and the given vectors are linearly independent on the interval (−∞,∞).

Want to see the full answer?

Chapter 9 Solutions

To prove: The determinant | t 2 t | t | 2 t 2 | t | | ≡ 0 and the given vectors are linearly independent on the interval ( − ∞ , ∞ ) .

Solution Summary: The author explains that the determinant left|cct2 and the given vectors are linearly independent on an interval.

Concept explainers

To prove: The determinant |t2t|t|2t2|t||≡0 and the given vectors are linearly independent on the interval (−∞,∞).

Want to see the full answer?

Chapter 9 Solutions

To prove:

The determinant |t2t|t|2t2|t||≡0 and the given vectors are linearly independent on the interval (−∞,∞).