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Consider the function F: RR defined by cos(xt) () == // F(x):= dr for xER. a) Show that F is well-defined and that F'(x), F"(x) exist for all x € R. b) Show that x²F"(x)+xF'(x)+x²F(x)=0 for all x € R.

Consider the function F: RR defined by cos(xt) () == // F(x):= dr for xER. a) Show that F is well-defined and that F'(x), F"(x) exist for all x € R. b) Show that x²F"(x)+xF'(x)+x²F(x)=0 for all x € R.

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 the function F: RR defined by cos(xt) F(x):= x):= [' dr for xER. a) Show that F is well-defined and that F'(x), F"(x) exist for all x € R. b) Show that x2²F"(x) +xF'(x)+x²F(x)=0 for all x = R. Hint: Partial integration of the expression obtained for F'(x) may help.
Transcribed Image Text:Consider the function F: RR defined by cos(xt) F(x):= x):= [' dr for xER. a) Show that F is well-defined and that F'(x), F"(x) exist for all x € R. b) Show that x2²F"(x) +xF'(x)+x²F(x)=0 for all x = R. Hint: Partial integration of the expression obtained for F'(x) may help.
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