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Find general solutions in powers of x of the differential equa- tions in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.

Find general solutions in powers of x of the differential equa- tions in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case.

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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Find general solutions in powers of \( x \) of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case. 1. \((x^2 - 1)y'' + 4xy' + 2y = 0\) 2. \((x^2 + 2)y'' + 4xy' + 2y = 0\) 3. \(y'' + xy' + y = 0\) 4. \((x^2 + 1)y'' + 6xy' + 4y = 0\) 5. \((x^2 - 3)y'' + 2xy' = 0\) 6. \((x^2 - 1)y'' - 6xy' + 12y = 0\) 7. \((x^2 + 3)y'' - 7xy' + 16y = 0\) 8. \((2 - x^2)y'' - xy' + 16y = 0\) 9. \((x^2 - 1)y'' + 8xy' + 12y = 0\) 10. \(3y'' + xy' - 4y = 0\) 11. \(5y'' - 2xy' + 10y = 0\) 12. \(y'' - x^2y' - 3xy = 0\) 13. \(y'' + x^2y' + 2xy = 0\) 14. \(y'' + xy = 0\) (an Airy equation) 15. \(y'' + x^2y = 0\)
Transcribed Image Text:Find general solutions in powers of \( x \) of the differential equations in Problems 1 through 15. State the recurrence relation and the guaranteed radius of convergence in each case. 1. \((x^2 - 1)y'' + 4xy' + 2y = 0\) 2. \((x^2 + 2)y'' + 4xy' + 2y = 0\) 3. \(y'' + xy' + y = 0\) 4. \((x^2 + 1)y'' + 6xy' + 4y = 0\) 5. \((x^2 - 3)y'' + 2xy' = 0\) 6. \((x^2 - 1)y'' - 6xy' + 12y = 0\) 7. \((x^2 + 3)y'' - 7xy' + 16y = 0\) 8. \((2 - x^2)y'' - xy' + 16y = 0\) 9. \((x^2 - 1)y'' + 8xy' + 12y = 0\) 10. \(3y'' + xy' - 4y = 0\) 11. \(5y'' - 2xy' + 10y = 0\) 12. \(y'' - x^2y' - 3xy = 0\) 13. \(y'' + x^2y' + 2xy = 0\) 14. \(y'' + xy = 0\) (an Airy equation) 15. \(y'' + x^2y = 0\)
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