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 ..

7th Edition

ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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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

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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,∞).

Want to see the full answer?

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

Concept explainers

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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,∞).