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

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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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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