To determine Whether the given functions are linearly dependent or linearly independent, if it is linearly dependent find the linear relation among them. Explanation Definition used: Linear dependence and independence: “The functions f 1 , f 2 , … , f n are said to be linearly dependent on an interval I if there exists a set of constants k 1 , k 2 , … , k n not all zero, such that k 1 f 1 ( t ) + k 2 f 2 ( t ) + ⋯ + k n f n ( t ) = 0 for all t in I . The functions f 1 , f 2 , … , f n are said to be linearly independent on I if they are not linearly dependent there.” Theorem used: If the function p 1 , p 2 , … , p n are continuous on the open interval I , if the functions y 1 , y 2 , … , y n are solutions of the equation L [ y ] = y ( n ) + p 1 ( t ) y ( n − 1 ) + ⋯ + p n − 1 ( t ) y ′ + p n ( t ) y = 0 and if W ( y 1 , y 2 , … , y n ) ( t ) ≠ 0 for at least one point in I , then every solution of the equation L [ y ] = y ( n ) + p 1 ( t ) y ( n − 1 ) + ⋯ + p n − 1 ( t ) y ′ + p n ( t ) y = 0 can be expressed as a linear combination of the solution y 1 , y 2 , … , y n . And Wronskian of the function is W ( y 1 , y 2 , … , y n ) = | y 1 y 2 ⋯ y n y 1 ′ y 2 ′ ⋯ y n ′ ⋮ ⋮ ⋮ y 1 ( n − 1 ) y 2 ( n − 1 ) ⋯ y n ( n − 1 ) | . Calculation: Consider the function f 1 ( t ) = 2 t − 3 , f 2 ( t ) = t 2 + 1 , f 3 ( t ) = 2 t 2 − t and f 4 ( t ) = t 2 + t + 1 . Wronskian of the above function can be calculated as follows. W ( f 1 , f 2 , f 3 , f 4 ) = | 2 t − 3 t 2 + 1 2 t 2 − t t 2 + t + 1 2 2 t 4 t − 1 2 t + 1 0 2 4 2 0 0 0 0 | = { ( 2 t − 3 ) | 2 t 4 t − 1 2 t + 1 2 4 2 0 0 0 | − ( t 2 + 1 ) | 2 4 t − 1 2 t + 1 0 4 2 0 0 0 | + ( 2 t 2 − t ) | 2 2 t 2 t + 1 0 2 2 0 0 0 | − ( t 2 + t + 1 ) | 2 2 t 4 t − 1 0 2 4 0 0 0 | } = { ( 2 t − 3 ) [ ( 2 t ) [ 0 − 0 ] − ( 4 t − 1 ) ( 0 − 0 ) + ( 2 t + 1 ) ( 0 − 0 ) ] − ( t 2 + 1 ) [ 2 ( 0 − 0 ) − ( 4 t − 1 ) ( 0 − 0 ) + ( 2 t + 1 ) ( 0 − 0 ) ] + ( 2 t 2 − t ) [ ( 2 ) ( 0 − 0 ) − 2 t ( 0 − 0 ) + ( 2 t + 1 ) ( 0 − 0 ) ] − ( t 2 + t + 1 ) [ ( 2 ) ( 0 − 0 ) − 2 t ( 0 − 0 ) + ( 4 t − 1 ) ( 0 − 0 ) ] } = 0 Hence, the Wronskian of the above functions W ( f 1 , f 2 , f 3 , f 4 ) = 0 Thus, the functions are linearly dependent

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ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 4.1, Problem 9P

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Whether the given functions are linearly dependent or linearly independent, if it is linearly dependent find the linear relation among them.

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Chapter 4.1, Problem 8P

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Elementary Differential Equations

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