6.2E: Series Solutions About Ordinary Points (Exercises)

In Exercises 1-11 find the series in \(x\) for the general solution.

1. \((1+x^2)y''+6xy'+6y=0\)

2. \((1+x^2)y''+2xy'-2y=0\)

3. \((1+x^2)y''-8xy'+20y=0\)

4. \((1-x^2)y''-8xy'-12y=0\)

5. \((1+x^2)y''-10xy'+28y=0\)

6. \(y''-xy=0\)

7. \(y''-2xy'+y=0\)

8. \(y''+x^2y'+xy=0\)

9. \((x-1)y''+y'=0\)

10. \(y''-(x+1)y'-y=0\)

11. \((x^2+2)y''+3xy'-y=0\)

In Exercises 12-23 find the power series solution \(y=\sum _{n=0}^{\infty} a_{n}x^{n}\) of the initial value problem.

12. \((1+x^2)y''+xy'+y=0,\quad y(0)=2,\quad y'(0)=-1\)

13. \((1+2x^2)y''-9xy'-6y=0,\quad y(0)=1,\quad y'(0)=-1\)

14. \((1+8x^2)y''+2y=0,\quad y(0)=2,\quad y'(0)=-1\)

15. \((1+3x)y''+xy'+2y=0,\quad y(0)=2,\quad y'(0)=-3\)

16. \((1+x+2x^2)y''+(2+8x)y'+4y=0,\quad y(0)=-1,\quad y'(0)=2\)

17. \((1-2x^2)y''+(2-6x)y'-2y=0,\quad y(0)=1,\quad y'(0)=0\)

18. \((1+x+3x^2)y''+(2+15x)y'+12y=0,\quad y(0)=0,\quad y'(0)=1\)

19. \((2+x)y''+(1+x)y'+3y=0,\quad y(0)=4,\quad y'(0)=3\)

20. \((3+3x+x^2)y''+(6+4x)y'+2y=0,\quad y(0)=7,\quad y'(0)=3\)

21. \((4+x)y''+(2+x)y'+2y=0,\quad y(0)=2,\quad y'(0)=5\)

22. \((x-1)y''-xy'+y=0,\quad y(0)=-2,\quad y'(0)=6\)

23. \(y''-2xy'+8y=0,\quad y(0)=3,\quad y'(0)=0\)

 

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