(a) Explanation 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: Let y 1 and y 2 be two solutions to the equation y ″ ( t ) + p y ′ ( t ) + q y ( t ) = 0 . Thus, the equation y ″ ( t ) + p y ′ ( t ) + q y ( t ) = 0 for y 1 ( t ) becomes, y ″ 1 ( t ) + p y ′ 1 ( t ) + q y 1 ( t ) = 0 y ″ 1 ( t ) = − p y ′ 1 ( t ) − q y 1 ( t ) Similarly, the equation y ″ ( t ) + p y ′ ( t ) + q y ( t ) = 0 for y 2 ( t ) becomes, y ″ 2 ( t ) + p y ′ 2 ( t ) + q y 2 ( t ) = 0 y ″ 2 ( t ) = − p y ′ 2 ( t ) − q y 2 ( t ) The Wronskian of y 1 and y 2 is given by W [ y 1 , y 2 ] ( t ) = y 1 ( t ) y ′ 2 ( t ) − y 2 ( t ) y ′ 1 ( t ) (b) To determine To solve: The separable equation W ′ + p W = 0 . (c) To determine To explain: The reason for how the Abel’s formula clarifies that Wronskian is either identically zero or never zero on ( a , b ) .

Student's Solutions Manual for Fundamentals of Differential Equations and Fundamentals of Differential Equations and Boundary Value Problems

9th Edition

ISBN: 9780321977212

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

Publisher: PEARSON

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Chapter 4.7, Problem 32E

(a)

To determine

To show: The Wronskian W satisfies the equation W′+pW=0.

(b)

To determine

To solve: The separable equation W′+pW=0.

(c)

To determine

To explain: The reason for how the Abel’s formula clarifies that Wronskian is either identically zero or never zero on (a,b).

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Chapter 4.7, Problem 31E

Chapter 4.7, Problem 33E

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The Wronskian W satisfies the equation W ′ + p W = 0 .

Solution Summary: The author explains that the Wronskian W satisfies the equation Wprime+pW=0.

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To show: The Wronskian W satisfies the equation W′+pW=0.

To solve: The separable equation W′+pW=0.

To explain: The reason for how the Abel’s formula clarifies that Wronskian is either identically zero or never zero on (a,b).

Want to see the full answer?

Chapter 4 Solutions