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Whether the functions e 2 x , x e 2 x , x 2 e 2 x linearly independent or dependent on the half-axis x ≥ 0 . The functions e 2 x , x e 2 x , x 2 e 2 x are linearly independent _ . Formula used: The Wronskian W of n solutions y 1 , ⋅ ⋅ ⋅ , y n defined as the n th − order determinant W ( y 1 , ⋅ ⋅ ⋅ , y n ) = | y 1 y 2 ⋅ ⋅ ⋅ y n y 1 ′ y 2 ′ ⋅ ⋅ ⋅ y 2 ′ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ y 1 ( n − 1 ) y 2 ( n − 1 ) ⋅ ⋅ ⋅ y n ( n − 1 ) | . Calculation: Suppose that the given functions are y 1 = e 2 x , y 2 = x e 2 x and y 3 = x 2 e 2 x . Differentiate the function y 1 = e 2 x , twice with respect to x . ( y 1 ) ″ = [ d 2 d x 2 ( e 2 x ) ] = [ 2 d d x ( e 2 x ) ] = 4 e 2 x Differentiate the function y 2 = x e 2 x , twice with respect to x . ( y 2 ) ″ = d d x [ d d x ( x e 2 x ) ] = d d x [ ( 2 x e 2 x + e 2 x ) ] = 2 e 2 x + 2 e 2 x + 4 x e 2 x = 4 e 2 x ( 1 + x ) Differentiate the function y 3 = x 2 e 2 x , twice with respect to x . ( y 3 ) ″ = d d x [ d d x ( x 2 e 2 x ) ] = d d x [ 2 x e 2 x + 2 x 2 e 2 x ] = 2 e 2 x + 4 x e 2 x + 4 x e 2 x + 4 x 2 e 2 x = 2 e 2 x ( 1 + 4 x + 2 x 2 ) The Wronskian W of the function for x ≥ 0 is calculated as: W ( y 1 , y 2 , y 3 ) | = | y 1 y 2 y 3 y 1 ′ y 2 ′ y 3 ′ y 1 ″ y 2 ″ y 3 ″ | = | e 2 x x e 2 x x 2 e 2 x 2 e 2 x e 2 x ( 1 + 2 x ) 2 x e 2 x ( 1 + x ) 4 e 2 x 4 e 2 x ( 1 + x ) 2 e 2 x ( 1 + 4 x + 2 x 2 ) | = e 2 x | 1 x x 2 2 ( 1 + 2 x ) 2 x ( 1 + x ) 4 4 ( 1 + x ) 2 ( 1 + 4 x + 2 x 2 ) | ( R 2 → R 3 − 2 R 2 R 4 → R 4 − 4 R 1 ) = e 2 x | 1 x x 2 0 1 x 0 4 2 + 8 x | On further simplification, the following obtained. W ( y 1 , y 2 , y 3 ) | = e 2 x ( 2 + 8 x − 4 x ) = e 2 x ( 2 + 4 x ) ≠ 0 From the above calculation, it can be observed that the value of Wronskian is nonzero. Note that, two functions are linearly dependent if their Wronskian is equal to zero and are linearly independent if their Wronskian is not equal to zero. Therefore, the functions e 2 x , x e 2 x , x 2 e 2 x are linearly independent _ .

Whether the functions e 2 x , x e 2 x , x 2 e 2 x linearly independent or dependent on the half-axis x ≥ 0 . The functions e 2 x , x e 2 x , x 2 e 2 x are linearly independent _ . Formula used: The Wronskian W of n solutions y 1 , ⋅ ⋅ ⋅ , y n defined as the n th − order determinant W ( y 1 , ⋅ ⋅ ⋅ , y n ) = | y 1 y 2 ⋅ ⋅ ⋅ y n y 1 ′ y 2 ′ ⋅ ⋅ ⋅ y 2 ′ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ y 1 ( n − 1 ) y 2 ( n − 1 ) ⋅ ⋅ ⋅ y n ( n − 1 ) | . Calculation: Suppose that the given functions are y 1 = e 2 x , y 2 = x e 2 x and y 3 = x 2 e 2 x . Differentiate the function y 1 = e 2 x , twice with respect to x . ( y 1 ) ″ = [ d 2 d x 2 ( e 2 x ) ] = [ 2 d d x ( e 2 x ) ] = 4 e 2 x Differentiate the function y 2 = x e 2 x , twice with respect to x . ( y 2 ) ″ = d d x [ d d x ( x e 2 x ) ] = d d x [ ( 2 x e 2 x + e 2 x ) ] = 2 e 2 x + 2 e 2 x + 4 x e 2 x = 4 e 2 x ( 1 + x ) Differentiate the function y 3 = x 2 e 2 x , twice with respect to x . ( y 3 ) ″ = d d x [ d d x ( x 2 e 2 x ) ] = d d x [ 2 x e 2 x + 2 x 2 e 2 x ] = 2 e 2 x + 4 x e 2 x + 4 x e 2 x + 4 x 2 e 2 x = 2 e 2 x ( 1 + 4 x + 2 x 2 ) The Wronskian W of the function for x ≥ 0 is calculated as: W ( y 1 , y 2 , y 3 ) | = | y 1 y 2 y 3 y 1 ′ y 2 ′ y 3 ′ y 1 ″ y 2 ″ y 3 ″ | = | e 2 x x e 2 x x 2 e 2 x 2 e 2 x e 2 x ( 1 + 2 x ) 2 x e 2 x ( 1 + x ) 4 e 2 x 4 e 2 x ( 1 + x ) 2 e 2 x ( 1 + 4 x + 2 x 2 ) | = e 2 x | 1 x x 2 2 ( 1 + 2 x ) 2 x ( 1 + x ) 4 4 ( 1 + x ) 2 ( 1 + 4 x + 2 x 2 ) | ( R 2 → R 3 − 2 R 2 R 4 → R 4 − 4 R 1 ) = e 2 x | 1 x x 2 0 1 x 0 4 2 + 8 x | On further simplification, the following obtained. W ( y 1 , y 2 , y 3 ) | = e 2 x ( 2 + 8 x − 4 x ) = e 2 x ( 2 + 4 x ) ≠ 0 From the above calculation, it can be observed that the value of Wronskian is nonzero. Note that, two functions are linearly dependent if their Wronskian is equal to zero and are linearly independent if their Wronskian is not equal to zero. Therefore, the functions e 2 x , x e 2 x , x 2 e 2 x are linearly independent _ .

Advanced Engineering Mathematics

Advanced Engineering Mathematics

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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)5. The operator T on a vector space V over Q has as its sequence of invariant factors the sequence (x-1)3(x-2)5(x-3)2, (x-1)(x-2)(x-3)2, (x-3). Find the Jordan normal form of T.

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Chapter 3 Solutions

Advanced Engineering Mathematics

Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P Ch. 3.1 - Prob. 11P

Ch. 3.1 - Prob. 12P Ch. 3.1 - Prob. 13P Ch. 3.1 - Prob. 14P Ch. 3.1 - Prob. 15P Ch. 3.1 - Prob. 16P Ch. 3.2 - Prob. 1P Ch. 3.2 - Prob. 2P Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - Prob. 8P Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - CAS EXPERIMENT. Reduction of Order. Starting with...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Prob. 5P Ch. 3.3 - Prob. 6P Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Prob. 10P Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3 - Prob. 1RQ Ch. 3 - List some other basic theorems that extend from...Ch. 3 - If you know a general solution of a homogeneous...Ch. 3 - What form does an initial value problem for an...Ch. 3 - What is the Wronskian? What is it used for? Ch. 3 - Prob. 6RQ Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Prob. 9RQ Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Prob. 12RQ Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Prob. 14RQ Ch. 3 - Prob. 15RQ Ch. 3 - Solve the IVP. Show the details of your work. Ch. 3 - Solve the IVP. Show the details of your work. y‴ +...Ch. 3 - Solve the IVP. Show the details of your work. Ch. 3 - Solve the IVP. Show the details of your work. Ch. 3 - Solve the IVP. Show the details of your work.

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