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5. (x2 - 3)y" + 2xy' = 0

5. (x2 - 3)y" + 2xy' = 0

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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**Title: Differential Equations and Power Series Solutions** **Introduction:** For problems 1 through 15, we aim to find the general solutions in powers of \( x \) for each given differential equation. Along with finding the solutions, we must state the recurrence relation and the guaranteed radius of convergence for each case. **Problems:** 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. \(y'' + x^2y = 0\) **Description:** These problems are designed to test understanding of solving second-order linear differential equations using power series methods. The task involves deriving a series solution, identifying the recurrence relations for coefficients, and ensuring that the solution converges within a specified radius from the point of expansion. Each equation increases in complexity, offering a range of contexts where different mathematical techniques may be required. **Notes:** Understanding the nature of each differential equation, whether it's homogeneous or non-homogeneous, and the possibility of singular points, is crucial
Transcribed Image Text:**Title: Differential Equations and Power Series Solutions** **Introduction:** For problems 1 through 15, we aim to find the general solutions in powers of \( x \) for each given differential equation. Along with finding the solutions, we must state the recurrence relation and the guaranteed radius of convergence for each case. **Problems:** 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. \(y'' + x^2y = 0\) **Description:** These problems are designed to test understanding of solving second-order linear differential equations using power series methods. The task involves deriving a series solution, identifying the recurrence relations for coefficients, and ensuring that the solution converges within a specified radius from the point of expansion. Each equation increases in complexity, offering a range of contexts where different mathematical techniques may be required. **Notes:** Understanding the nature of each differential equation, whether it's homogeneous or non-homogeneous, and the possibility of singular points, is crucial
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