Explanation Given: The functions are f 1 ( x ) = e 2 x , f 2 ( x ) = e 3 x , f 3 ( x ) = e − x . Approach: The functions f 1 ( x ) , f 2 ( x ) , ... , f n ( x ) are said to be linearly dependent if | f 1 ( x ) f 2 ( x ) ... f n ( x ) f 1 ′ ( x ) f 2 ′ ( x ) ... f n ′ ( x ) ... ... ... ... f 1 ( n − 1 ) ( x ) f 2 ( n − 1 ) ( x ) ... f n ( n − 1 ) ( x ) | = 0 , otherwise functions are linearly independent, Where f 1 ( n − 1 ) ( x ) denotes d n − 1 d x n − 1 ( f 1 ( x ) ) . Calculation: Compute the first derivative of the functions f 1 ( x ) = e 2 x , f 2 ( x ) = e 3 x , f 3 ( x ) = e − x as follows: f 1 ′ ( x ) = d d x ( e 2 x ) , f 2 ′ ( x ) = d d x ( e 3 x ) , f 3 ′ ( x ) = d d x ( e − x ) . Differentiate using power rule, then f 1 ′ ( x ) = 2 e 2 x , f 2 ′ ( x ) = 3 e 3 x , f 3 ′ ( x ) = − e − x . Compute the second derivative of the functions as follows: f 1 ′ ′ ( x ) = d 2 d x 2 ( e 2 x ) , f 2 ′ ′ ( x ) = d 2 d x 2 ( e 3 x ) , f 3 ′ ′ ( x ) = d 2 d x 2 ( e − x ) Differentiate using power rule, then f 1 ′ ′ ( x ) = 4 e 2 x , f 2 ′ ′ ( x ) = 9 e 3 x , f 3 ′ ′ ( x ) = e − x

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Question

Chapter 4.5, Problem 35P

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To show:

The functions f1(x)=e2x,f2(x)=e3x,f3(x)=e−x are linearly independent on interval (−∞,∞) using Wronskian method.

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Chapter 4.5, Problem 33P

Chapter 4.5, Problem 36P

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Chapter 4 Solutions

Differential Equations and Linear Algebra (4th Edition)

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The functions f 1 ( x ) = e 2 x , f 2 ( x ) = e 3 x , f 3 ( x ) = e − x are linearly independent on interval ( − ∞ , ∞ ) using Wronskian method.

To show: The functions f1(x)=e2x,f2(x)=e3x,f3(x)=e−x are linearly independent on interval (−∞,∞) using Wronskian method.

Want to see the full answer?

Chapter 4 Solutions

To show:

The functions f1(x)=e2x,f2(x)=e3x,f3(x)=e−x are linearly independent on interval (−∞,∞) using Wronskian method.