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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. 1. (x2 - 1)y" + 4xy' + 2y = 0 2. (x2 + 2)y" + 4xy' + 2y 0 3. y" + xy' + y = 0 4. (x2 + 1)y" + 6xy' +4y 0 5. (x2 - 3)y" + 2xy' 0 7. (x+3)y 8. (2 x)y" - xy' + 16y = 0 9. (x2 – 1)y" + 8xy+ 12y =0 10. 3y" + xy' – 4y = 0 11. 5y" – 2xy + 10y = 0 12. y" – x?y' – 3xy = 0 13. y" +x'y' + 2ry 0 14. y" + xy = 0 (an Airy equation) 15. y" +x²y = 0 %3D | Exu' t 12y = 0

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. 1. (x2 - 1)y" + 4xy' + 2y = 0 2. (x2 + 2)y" + 4xy' + 2y 0 3. y" + xy' + y = 0 4. (x2 + 1)y" + 6xy' +4y 0 5. (x2 - 3)y" + 2xy' 0 7. (x+3)y 8. (2 x)y" - xy' + 16y = 0 9. (x2 – 1)y" + 8xy+ 12y =0 10. 3y" + xy' – 4y = 0 11. 5y" – 2xy + 10y = 0 12. y" – x?y' – 3xy = 0 13. y" +x'y' + 2ry 0 14. y" + xy = 0 (an Airy equation) 15. y" +x²y = 0 %3D | Exu' t 12y = 0

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**Title: Solving Differential Equations in Power Series** **Objective:** Find general solutions in powers of \(x\) for the differential equations listed below. For each problem, determine the recurrence relation and the guaranteed radius of convergence. **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. \(y'' + xy = 0 \quad (\text{an Airy equation})\) 15. \(y'' + x^2y = 0\) **Instructions:** - Determine the power series solution for each equation. - Establish the recurrence relation for the series coefficients. - Calculate the radius of convergence for each series solution. **Graphical Representation:** There are no graphs or diagrams included in this problem set. Each equation will require analytical solutions and calculations based on series expansions. **Note:** These exercises focus on developing skills in solving differential equations using power series methods, and understanding solution behavior in terms of convergence and applicable boundaries.
Transcribed Image Text:**Title: Solving Differential Equations in Power Series** **Objective:** Find general solutions in powers of \(x\) for the differential equations listed below. For each problem, determine the recurrence relation and the guaranteed radius of convergence. **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. \(y'' + xy = 0 \quad (\text{an Airy equation})\) 15. \(y'' + x^2y = 0\) **Instructions:** - Determine the power series solution for each equation. - Establish the recurrence relation for the series coefficients. - Calculate the radius of convergence for each series solution. **Graphical Representation:** There are no graphs or diagrams included in this problem set. Each equation will require analytical solutions and calculations based on series expansions. **Note:** These exercises focus on developing skills in solving differential equations using power series methods, and understanding solution behavior in terms of convergence and applicable boundaries.
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