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Math Advanced Math ADVANCED ENGINEERING MATHEMATICS (LL)

Whether the functions cos 2 x , sin 2 x , 2 π linearly independent or dependent on the half-axis x ≥ 0 . The functions cos 2 x , sin 2 x , 2 π are linearly dependent _ . 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 = cos 2 x , y 2 = sin 2 x and y 3 = 2 π . Differentiate the function y 1 = cos 2 x , twice with respect to x . ( y 1 ) ″ = d d x [ d d x ( cos 2 x ) ] = d d x [ − ( 2 cos x sin x ) ] = d d x [ − ( sin 2 x ) ] = − 2 cos 2 x Differentiate the function y 2 = sin 2 x , twice with respect to x . ( y 2 ) ″ = [ d 2 d x 2 ( sin 2 x ) ] = [ d d x ( 2 sin x cos x ) ] = [ d d x ( sin 2 x ) ] = 2 cos 2 x Differentiate the function y 3 = 2 π , twice with respect to x . ( y 3 ) ″ = d d x [ d d x ( 2 π ) ] = d d x [ 0 ] = 0 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 ″ | = | sin 2 x cos 2 x 2 π sin 2 x − sin 2 x 0 2 cos 2 x − 2 cos 2 x 0 | = 2 π ( sin 2 x × − 2 cos 2 x + 2 cos 2 x × sin 2 x ) = 0 From the above calculation, it can be observed that the value of value of Wronskian W is zero. 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 cos 2 x , sin 2 x , 2 π are linearly dependent _ .

Whether the functions cos 2 x , sin 2 x , 2 π linearly independent or dependent on the half-axis x ≥ 0 . The functions cos 2 x , sin 2 x , 2 π are linearly dependent _ . 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 = cos 2 x , y 2 = sin 2 x and y 3 = 2 π . Differentiate the function y 1 = cos 2 x , twice with respect to x . ( y 1 ) ″ = d d x [ d d x ( cos 2 x ) ] = d d x [ − ( 2 cos x sin x ) ] = d d x [ − ( sin 2 x ) ] = − 2 cos 2 x Differentiate the function y 2 = sin 2 x , twice with respect to x . ( y 2 ) ″ = [ d 2 d x 2 ( sin 2 x ) ] = [ d d x ( 2 sin x cos x ) ] = [ d d x ( sin 2 x ) ] = 2 cos 2 x Differentiate the function y 3 = 2 π , twice with respect to x . ( y 3 ) ″ = d d x [ d d x ( 2 π ) ] = d d x [ 0 ] = 0 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 ″ | = | sin 2 x cos 2 x 2 π sin 2 x − sin 2 x 0 2 cos 2 x − 2 cos 2 x 0 | = 2 π ( sin 2 x × − 2 cos 2 x + 2 cos 2 x × sin 2 x ) = 0 From the above calculation, it can be observed that the value of value of Wronskian W is zero. 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 cos 2 x , sin 2 x , 2 π are linearly dependent _ .

ADVANCED ENGINEERING MATHEMATICS (LL)

ADVANCED ENGINEERING MATHEMATICS (LL)

10th Edition

ISBN: 9781119455929

Author: Kreyszig

Publisher: WILEY

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Chapter 3 Solutions

ADVANCED ENGINEERING MATHEMATICS (LL)

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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