Explanation Given: A second-order equation is said to be in Sturm-Liouville form if it is expressed as [ p ( t ) y ′ ( t ) ] ′ + q ( t ) y ( t ) = 0 . y ( 0 ) = a , and y ′ ( 0 ) = b are the initial values for the Sturm-Liouville problem Approach: Assume, x 1 = y x 2 = p y ′ ( 1 ) Express Sturm-Liouville as [ p ( t ) y ′ ( t ) ] ′ + q ( t ) y ( t ) = 0 [ p ( t ) y ′ ( t ) ] ′ = − q ( t ) y ( t ) ( 2 ) Calculation: From equation ( 1 ) x 1 = y x ′ 1 = y ′ x 2 = p y ′ x 2 = p x ′ 1 ( 3 ) Do substitution in above equation from equation ( 1 ) and equation ( 2 ) . [ p ( t ) y ′ ( t ) ] ′ = − q ( t ) y ( t ) x ′ 2 = − q x 1 ( 4 ) From equation ( 3 ) and equation ( 4 )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 5.3, Problem 8E

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To show:

The substitutions x1=y, x2=py′ result in the normal form

x′1=x2/px′2=−qx1.

And find the values of x1(0) and x2(0).

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Chapter 5.3, Problem 7E

Chapter 5.3, Problem 9E

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Fundamentals of Differential Equations and Boundary Value Problems

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The substitutions x 1 = y , x 2 = p y ′ result in the normal form x ′ 1 = x 2 / p x ′ 2 = − q x 1 . And find the values of x 1 ( 0 ) and x 2 ( 0 ) .

Solution Summary: The author explains that a second-order equation is said to be in Sturm-Liouville form if it is expressed as rll

Concept explainers

Videos

To show:

The substitutions x1=y, x2=py′ result in the normal form

x′1=x2/px′2=−qx1.

And find the values of x1(0) and x2(0).

Want to see the full answer?

Chapter 5 Solutions

The substitutions x 1 = y , x 2 = p y ′ result in the normal form x ′ 1 = x 2 / p x ′ 2 = − q x 1 . And find the values of x 1 ( 0 ) and x 2 ( 0 ) .

Solution Summary: The author explains that a second-order equation is said to be in Sturm-Liouville form if it is expressed as rll

Concept explainers

Videos

To show: The substitutions x1=y, x2=py′ result in the normal form x′1=x2/px′2=−qx1. And find the values of x1(0) and x2(0).

Want to see the full answer?

Chapter 5 Solutions

To show:

The substitutions x1=y, x2=py′ result in the normal form

x′1=x2/px′2=−qx1.

And find the values of x1(0) and x2(0).