Explanation Given: The relation between the two functions are given by, c 1 y 1 ( t ) + c 2 y 2 ( t ) = 0 …… ( 1 ) The interval in which the linear dependency is to be checked is ( 0 , 1 ) . Approach: If one function is the constant multiple of another in a certain interval then both functions will be linearly dependent to each other in that particular interval. The relation between the function which are linearly dependent is, y 1 ( t ) = k y 2 ( t ) …… ( 2 ) Here, the constant is k . Calculation: Substitute k y 2 ( t ) for y 1 ( t ) in the equation ( 1 ) . c 1 k y 2 ( t ) + c 2 y 2 ( t ) = 0 ( c 1 k + c 2 ) y 2 ( t ) = 0 From the above equation. c 1 k + c 2 = 0 k = − k c 1 To satisfy the criteria for the linear dependency of two functions it is not compulsory to have a value of zero for constant c 1 and c 2 . Hence, if the two functions y 1 and y 2 are linearly dependent then there exists some value by the constants

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 4.2, Problem 33E

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Whether two functions are linearly dependent on the interval I if and only if there exist constants c1 and c2, not both zero, or not.

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Chapter 4.2, Problem 32E

Chapter 4.2, Problem 34E

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Whether two functions are linearly dependent on the interval I if and only if there exist constants c 1 and c 2 , not both zero, or not.

Solution Summary: The author proves that two functions are linearly dependent on the interval I if there exist constants c_1 and

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To find:

Whether two functions are linearly dependent on the interval I if and only if there exist constants c1 and c2, not both zero, or not.

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Chapter 4 Solutions

Whether two functions are linearly dependent on the interval I if and only if there exist constants c 1 and c 2 , not both zero, or not.

Solution Summary: The author proves that two functions are linearly dependent on the interval I if there exist constants c_1 and

Videos

To find: Whether two functions are linearly dependent on the interval I if and only if there exist constants c1 and c2, not both zero, or not.

Want to see the full answer?

Chapter 4 Solutions

To find:

Whether two functions are linearly dependent on the interval I if and only if there exist constants c1 and c2, not both zero, or not.