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Consider y"(x) + r(x) y'(x) + s(x) y(x) = 0. Its "normal form" DE is u"(x) + S(x) u(x) = 0, where S(x) = s(x) - -r(x)? r'(x). The nontrivial solutions y(x) and u(x), if oscillatory, must have the same zeros. (a) Find the normal form of Bessel's DE of order 1/3. (b) By referring to the Sturm comparison theorem and a suitable well-known DE,, explain why the separation of the zeros of the Bessel functions J(x) and Y(x) must be less than î. (c) What does the Sturm separation theorem conclude regarding the zeros of J3(x) and Y3(x) ?

Consider y"(x) + r(x) y'(x) + s(x) y(x) = 0. Its "normal form" DE is u"(x) + S(x) u(x) = 0, where S(x) = s(x) - -r(x)? r'(x). The nontrivial solutions y(x) and u(x), if oscillatory, must have the same zeros. (a) Find the normal form of Bessel's DE of order 1/3. (b) By referring to the Sturm comparison theorem and a suitable well-known DE,, explain why the separation of the zeros of the Bessel functions J(x) and Y(x) must be less than î. (c) What does the Sturm separation theorem conclude regarding the zeros of J3(x) and Y3(x) ?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider y"(x) + r(x) y'(x) + s(x) y(x) = 0. Its "normal form" 1 %3D DE is u"(x) + S(x) u(x) = 0, where S(x) = s(x) - r(x)² - r"(x) . The nontrivial solutions y(x) and u(x), if oscillatory, must have the same zeros. (a) Find the normal form of Bessel's DE of order 1/3. (b) By referring to the Sturm comparison theorem and a suitable well-known DE,, explain why the separation of the zeros of the Bessel functions J3(x) and Y3(x) must be less than 7. (c) What does the Sturm separation theorem conclude regarding the zeros of J3(x) and Y/3 (x) ?
Transcribed Image Text:Consider y"(x) + r(x) y'(x) + s(x) y(x) = 0. Its "normal form" 1 %3D DE is u"(x) + S(x) u(x) = 0, where S(x) = s(x) - r(x)² - r"(x) . The nontrivial solutions y(x) and u(x), if oscillatory, must have the same zeros. (a) Find the normal form of Bessel's DE of order 1/3. (b) By referring to the Sturm comparison theorem and a suitable well-known DE,, explain why the separation of the zeros of the Bessel functions J3(x) and Y3(x) must be less than 7. (c) What does the Sturm separation theorem conclude regarding the zeros of J3(x) and Y/3 (x) ?
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