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Math Advanced Math Fundamentals of Differential Equations (9th Edition)

The equation ( y f ) ′ = W [ f , y ] f 2 holds, where f ( t ) is the non-trivial solution to y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 and y ( t ) is the second linearly independent solution. Definition used: The Wronskian of y 1 ( t ) and y 2 ( t ) , denoted as W [ y 1 , y 2 ] is, W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y ′ 2 ( t ) − y 2 ( t ) y ′ 1 ( t ) = | y 1 ( t ) y 2 ( t ) y ′ 1 ( t ) y ′ 2 ( t ) | . Description: Given that, f ( t ) is the non-trivial solution to y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 and y ( t ) is the second linearly independent solution. That is, y 1 ( t ) = y ( t ) and y 2 ( t ) = f ( t ) . By the above mentioned definition, W [ f , y ] ( t ) = f ( t ) y ′ ( t ) − y ( t ) f ′ ( t ) . The value of ( y f ) ′ is obtained as follows: ( y f ) ′ = f y ′ − y f ′ f 2 = W [ f , y ] f 2 ( ∵ W [ f , y ] = f y ′ − y f ′ ) Therefore, ( y f ) ′ = W [ f , y ] f 2 . To prove: The reduction of order formula using the Abel’s identity for Wronskian in ( y f ) ′ = W [ f , y ] f 2 . Theorem used: Reduction of order: If y 1 ( t ) is a solution, not identically zero, to the homogenous differential equation y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 in an interval I , then y 2 ( t ) = y 1 ( t ) ∫ e − ∫ p ( t ) d t y 1 ( t ) 2 d t is a second, linearly independent solution. Description: If f ( t ) is the non-trivial solution to y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 and y ( t ) is the second linearly independent solution, then ( y f ) ′ = W [ f , y ] f 2 . Integrating on both sides as, ∫ ( y f ) ′ d t = ∫ W [ f , y ] f 2 d t y f = ∫ W [ f , y ] f 2 d t The Abel’s identity for Wronskian is, W = C e − ∫ p ( t ) d t . Thus, the equation y f = ∫ W [ f , y ] f 2 d t becomes, y f = ∫ C e − ∫ p ( t ) d t f 2 d t y = f ∫ C e − ∫ p ( t ) d t f 2 d t Replacing, the initial assumption y 1 ( t ) = y ( t ) and y 2 ( t ) = f ( t ) as y 2 ( t ) = y 1 ( t ) ∫ e − ∫ p ( t ) d t y 1 ( t ) 2 d t , which is same as the reduction of order formula. The above equation is used to find the second linearly independent solution y 2 ( t ) , if y 1 ( t ) is a solution to the homogenous differential equation y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 .

The equation ( y f ) ′ = W [ f , y ] f 2 holds, where f ( t ) is the non-trivial solution to y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 and y ( t ) is the second linearly independent solution. Definition used: The Wronskian of y 1 ( t ) and y 2 ( t ) , denoted as W [ y 1 , y 2 ] is, W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y ′ 2 ( t ) − y 2 ( t ) y ′ 1 ( t ) = | y 1 ( t ) y 2 ( t ) y ′ 1 ( t ) y ′ 2 ( t ) | . Description: Given that, f ( t ) is the non-trivial solution to y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 and y ( t ) is the second linearly independent solution. That is, y 1 ( t ) = y ( t ) and y 2 ( t ) = f ( t ) . By the above mentioned definition, W [ f , y ] ( t ) = f ( t ) y ′ ( t ) − y ( t ) f ′ ( t ) . The value of ( y f ) ′ is obtained as follows: ( y f ) ′ = f y ′ − y f ′ f 2 = W [ f , y ] f 2 ( ∵ W [ f , y ] = f y ′ − y f ′ ) Therefore, ( y f ) ′ = W [ f , y ] f 2 . To prove: The reduction of order formula using the Abel’s identity for Wronskian in ( y f ) ′ = W [ f , y ] f 2 . Theorem used: Reduction of order: If y 1 ( t ) is a solution, not identically zero, to the homogenous differential equation y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 in an interval I , then y 2 ( t ) = y 1 ( t ) ∫ e − ∫ p ( t ) d t y 1 ( t ) 2 d t is a second, linearly independent solution. Description: If f ( t ) is the non-trivial solution to y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 and y ( t ) is the second linearly independent solution, then ( y f ) ′ = W [ f , y ] f 2 . Integrating on both sides as, ∫ ( y f ) ′ d t = ∫ W [ f , y ] f 2 d t y f = ∫ W [ f , y ] f 2 d t The Abel’s identity for Wronskian is, W = C e − ∫ p ( t ) d t . Thus, the equation y f = ∫ W [ f , y ] f 2 d t becomes, y f = ∫ C e − ∫ p ( t ) d t f 2 d t y = f ∫ C e − ∫ p ( t ) d t f 2 d t Replacing, the initial assumption y 1 ( t ) = y ( t ) and y 2 ( t ) = f ( t ) as y 2 ( t ) = y 1 ( t ) ∫ e − ∫ p ( t ) d t y 1 ( t ) 2 d t , which is same as the reduction of order formula. The above equation is used to find the second linearly independent solution y 2 ( t ) , if y 1 ( t ) is a solution to the homogenous differential equation y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y ( t ) = 0 .

Solution Summary: The author explains that the Wronskian equation f(t) is the non-trivial solution to the two linearly independent solutions.

Fundamentals of Differential Equations (9th Edition)

Fundamentals of Differential Equations (9th Edition)

9th Edition

ISBN: 9780321977069

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 4.7, Problem 51E

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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 4 Solutions

Fundamentals of Differential Equations (9th Edition)

Ch. 4.1 - Verify that for b = 0 and Fext(t) = 0, equation...Ch. 4.1 - If Fext(t) = 0, equation (3) becomes my + by + ky...Ch. 4.1 - Show that if Fext(t) = 0, m = 1, k = 9, and b = 6,...Ch. 4.1 - Verify that y = sin 3t + 2 cos 3t is a solution to...Ch. 4.1 - Prob. 5E Ch. 4.1 - An external force F(t) = 2 cos 2t is applied to a...Ch. 4.1 - Prob. 7E Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Prob. 10E

Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 3E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - Prob. 10E Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 112, find a general solution to the...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 15E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 17E Ch. 4.2 - Prob. 18E Ch. 4.2 - Prob. 19E Ch. 4.2 - In Problems 1320, solve the given initial value...Ch. 4.2 - Prob. 21E Ch. 4.2 - Prob. 22E Ch. 4.2 - Prob. 23E Ch. 4.2 - Prob. 24E Ch. 4.2 - Prob. 25E Ch. 4.2 - Prob. 26E Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - In Problems 2732, use Definition 1 to determine...Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Prob. 33E Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Using the definition in Problem 35, prove that if...Ch. 4.2 - Prob. 37E Ch. 4.2 - Prob. 38E Ch. 4.2 - Prob. 39E Ch. 4.2 - Prob. 40E Ch. 4.2 - In Problems 3741, find three linearly independent...Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 43E Ch. 4.2 - Solve the initial value problem: y 2y y + 2y =...Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - Prob. 5E Ch. 4.3 - Prob. 6E Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 18, the auxiliary equation for the...Ch. 4.3 - In Problems 920, find a general solution. 9. y 8y...Ch. 4.3 - In Problems 920, find a general solution. 10. y +...Ch. 4.3 - Prob. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - In Problems 920, find a general solution. 13. y ...Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.3 - Prob. 16E Ch. 4.3 - In Problems 920, find a general solution. 17. y y...Ch. 4.3 - In Problems 920, find a general solution. 18. 2y +...Ch. 4.3 - Prob. 19E Ch. 4.3 - In Problems 920, find a general solution. 20. y y...Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 22E Ch. 4.3 - Prob. 23E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 25E Ch. 4.3 - In Problems 2127, solve the given initial value...Ch. 4.3 - Prob. 27E Ch. 4.3 - Prob. 28E Ch. 4.3 - Prob. 29E Ch. 4.3 - Prob. 30E Ch. 4.3 - Prob. 31E Ch. 4.3 - Prob. 32E Ch. 4.3 - Prob. 33E Ch. 4.3 - Prob. 34E Ch. 4.3 - Swinging Door. 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