Explanation Given: The equations is y ″ + ( x 2 − 1 ) y = 0 Approach: 1) Suppose that ϕ 1 and ϕ 2 are nontrivial solutions to the differential equations y ″ + Q 1 ( x ) y = 0 y ″ + Q 2 ( x ) y = 0 respectively on ( a , b ) . If Q 2 ( x ) ≥ Q 1 ( x ) for x in ( a , b ) then between any two consecutive zeroes x 1 and x 2 of ϕ 1 there exists a zero of ϕ 2 unless Q 2 ( x ) = Q 1 ( x ) on [ x 1 , x 2 ] and ϕ 1 and ϕ 2 are linearly dependent on [ x 1 , x 2 ] . 2) If Q ( x ) ≤ 0 on [ a , b ] , then a nontrivial solution to y ″ + Q ( x ) y = 0 has at most one zero in [ a , b ] . Calculation: Consider the equations y ″ + y = 0 ..

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 11.8, Problem 6E

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To prove:

A nontrivial solution to y″+(x2−1)y=0 has at most one zero in (−1,1) and has infinitely many zeroes in (−∞,−1) and (1,∞).

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Chapter 11.8, Problem 4E

Chapter 11.8, Problem 7E

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Chapter 11 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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To prove: A nontrivial solution to y ″ + ( x 2 − 1 ) y = 0 has at most one zero in ( − 1 , 1 ) and has infinitely many zeroes in ( − ∞ , − 1 ) and ( 1 , ∞ ) .

Solution Summary: The author explains that a nontrivial solution to y′′+(x2-1)y=0 has at most one zero and infinitely many zeroes

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To prove:

A nontrivial solution to y″+(x2−1)y=0 has at most one zero in (−1,1) and has infinitely many zeroes in (−∞,−1) and (1,∞).

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Chapter 11 Solutions

To prove: A nontrivial solution to y ″ + ( x 2 − 1 ) y = 0 has at most one zero in ( − 1 , 1 ) and has infinitely many zeroes in ( − ∞ , − 1 ) and ( 1 , ∞ ) .

Solution Summary: The author explains that a nontrivial solution to y′′+(x2-1)y=0 has at most one zero and infinitely many zeroes

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To prove: A nontrivial solution to y″+(x2−1)y=0 has at most one zero in (−1,1) and has infinitely many zeroes in (−∞,−1) and (1,∞).

Want to see the full answer?

Chapter 11 Solutions

To prove:

A nontrivial solution to y″+(x2−1)y=0 has at most one zero in (−1,1) and has infinitely many zeroes in (−∞,−1) and (1,∞).