# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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PROBLEMS In each of Problems 1 through 14 solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution.

1. y" 1 2. y" - xyf - 0

y II

II Î

,0 x

x 0,

î II

II y

0

3. y" 0 1 4. y" + k2 x 2y I 0, x0 I 0, k a constant

x

î"

è

y

1

x

³

5. (1 - x)y" + y = 0, x0 I0 6. (2 + x 2)y" - xy + 4y = 0, x0 = 0

7. y " + xy' + 2y = 0, x00 I0 8. xy "+ y' + II

Î

x

0,

II

y

x

9. (1 + x2)y" - 4xy' + 6y : I 0, x0 I 0 10. (4 2) +

x 2

1 y

II

0,

x

Î

I

0

11. (3 - x2)y" - 3xy; - y = 0, x0 0I 0 12. (1 - x)y" + xy' - y = 0, x00 = 0

13. 2y " + xy' + 3y = 0, x 0 I 0 0 14. 2y + 2

x II

+ 0

0x

0,

II

y

3

+

248

Chapter 5. Series Solutions of Second Order Linear Equations

In each of Problems 15 through 18:

(a) Find the first five nonzero terms in the solution of the given initial value problem.

(b) Plot the four-term and the five-term approximations to the solution on the same axes.

(c) From the plot in part (b) estimate the interval in which the four-term approximation is reasonably accurate.

> 15. y" — xy — y = 0, y(0) = 2, y (0) = 1; see Problem 2

> 16. (2 + x2)y" — xy' + 4y = 0, y(0) = —1, y'(0) = 3; see Problem 6

> 17. y" + xy' + 2y = 0, y(0) = 4, y'(0) = —1; see Problem 7

> 18. (1 — x)y" + xy1 — y = 0, y(0) = —3, y'(0) = 2; see Problem 12

19. By making the change of variable x — 1 = t and assuming that y is a power series in t, find two linearly independent series solutions of

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