7. y" + xy' + 2y = 0, xo = 0 11
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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Question
7 please write cle
Transcribed Image Text:Problems
In each of Problems 1 through 11:
a. Seek power series solutions of the given differential equation
about the given point xo; find the recurrence relation that the
coefficients must satisfy.
b. Find the first four nonzero terms in each of two solutions y₁
and y2 (unless the series terminates sooner).
c. By evaluating the Wronskian W[y1, y21(xo), show that yı
and y2 form a fundamental set of solutions.
d. If possible, find the general term in each solution.
-
1. y" - y = 0,
2. y" + 3y' = 0,
3.
4.
5.
6. (1-x)y"+y=0,
7. y" + xy' + 2y = 0,
8. xy"+y' + xy = 0,
9.
Xo = 0
SRS
xo = 0
y" - xy' - y = 0,
y" - xy' - y = 0,
y" +k²x²y = 0,
Xo = 0.
Xo = 1
xo = 0, k a constant
xo = 0
xo = 0
хо
x₁ = 1
(3-x²) y" - 3xy' - y = 0, xo = 0
10. 2y" + xy' + 3y = 0, xo = 0
11. 2y" + (x + 1) y' + 3y = 0, Xo = 2
In each of Problems 12 through 14:
a. Find the first five nonzero terms in the solution of the given
initial-value problem.
G 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.
12. y" - xy' - y = 0, y(0) = 2, y'(0) = 1; see Problem 3
13. y" + xy' +2y = 0, y(0) = 4, y'(0) = -1; see Problem 7
14. (1-x)y" + xy' - y = 0, y(0) = -3, y'(0) = 2
15. a. By making the change of variable x - 1 = t and assuming
that y has a Taylor series in powers of t, find two series solutions
of
y" + (x - 1)²y' + (x² - 1) y = 0
in powers of x - 1.
b. Show that you obtain the same result by assuming that y
coefficient x²-1 in
has a Taylor series in powers of x - 1 and also expressing the
1 in powers of x - 1.
16. Prove equation (10).
1.1
17. Show directly, using t
of Airy's equation about x =
of the text.
18. The Hermite Equatio
y" - 2xy' +
where is a constant, is k
important equation in mathe
a. Find the first four
= 0 and sho
about x
solutions.
b. Observe that if X
or the other of the ser
polynomial. Find the p
8, and 10. Note that ea
multiplicative constant.
c. The Hermite polyn
solution of the Hermit
coefficient of x" is 2".
19. Consider the initial-va
a. Show that y = s
problem.
b. Look for a solution
a power series about x
in x3 in this series.
In each of Problems 20 thr
series solution of the giver
thereby obtaining graphs ana
5.2.4 (except that we do not
solution).
G 20. y" + xy' + 2y =
G 21. (4-x2) y" + 2y
G 22. y" + x²y = 0,
G 23. (1-x) y" + xy'
5 Charles Hermite (1822-1901
algebraist. An inspiring teacher,
and the Sorbonne. He introduced
1873 that e is a transcendental nu
equation with rational coefficient
matrices (see Section 7.3), some
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