Xo = 1 5. y" +k²x²y = 0, xo = 0, k a constant
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
5 pl
, show that yı
and y2 form a fundamental set of solutions.
d. If possible, find the general term in each solution.
y" - y = 0,
xo = 0
y" + 3y' = 0,
y" -xy' - y = 0,
y" - xy' - y = 0,
y" +k²x²y = 0,
(1-x)y"+y=0,
xo = 0, ka constant
x=0
7.
y" + xy' + 2y = 0,
xo=0
8.
xy"+y' + xy = 0,
хо
x₁ = 1
9.
(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
ܚܕ ܕ ܚܕܐ ܚܕ ܗܐ ܗ ܝ
6.
xo = 0
section.
xo = 0
xo = 1
16. Prove equation (10).
PLE
y" + (x − 1)²y' + (x² - 1) y = 0
(1-1)
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.
3
1-1
17. Show directly, u
of Airy's equation abo
of the text.
18. The Hermite Ec
y" — 2x
-
where λ is a constant
important equation in r
a. Find the first
about x = 0 an
solutions.
19.
b. Observe that
or the other of th
polynomial. Find
8, and 10. Note th
multiplicative con
c. The Hermite p
solution of the He
coefficient of x" is
Consider the initi:
a. Show that y
problem.
b. Look for a solu
a power series abo
in x3 in this series.
In each of Problems 20
series solution of the
thereby obtaining graph
5.2.4 (except that we do
solution).
G 20. y" + xy' +2.
G 21. (4-x2)y" +
G 22. y" + x²y = 0
G 23. (1-x) y" +=
5Charles Hermite (1822-1
algebraist. An inspiring teac
and the Sorbonne. He introdu
1873 that e is a transcendenta
equation with rational coeffic
matrices (see Serti](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22b1319c-242c-4915-8f56-7c035e43f773%2Fd8cba5e2-eebd-4f02-8d8f-522564cb18a1%2Fs6eiuyt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
2.
-3.
4.
5.
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, y2](xo), show that yı
and y2 form a fundamental set of solutions.
d. If possible, find the general term in each solution.
y" - y = 0,
xo = 0
y" + 3y' = 0,
y" -xy' - y = 0,
y" - xy' - y = 0,
y" +k²x²y = 0,
(1-x)y"+y=0,
xo = 0, ka constant
x=0
7.
y" + xy' + 2y = 0,
xo=0
8.
xy"+y' + xy = 0,
хо
x₁ = 1
9.
(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
ܚܕ ܕ ܚܕܐ ܚܕ ܗܐ ܗ ܝ
6.
xo = 0
section.
xo = 0
xo = 1
16. Prove equation (10).
PLE
y" + (x − 1)²y' + (x² - 1) y = 0
(1-1)
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.
3
1-1
17. Show directly, u
of Airy's equation abo
of the text.
18. The Hermite Ec
y" — 2x
-
where λ is a constant
important equation in r
a. Find the first
about x = 0 an
solutions.
19.
b. Observe that
or the other of th
polynomial. Find
8, and 10. Note th
multiplicative con
c. The Hermite p
solution of the He
coefficient of x" is
Consider the initi:
a. Show that y
problem.
b. Look for a solu
a power series abo
in x3 in this series.
In each of Problems 20
series solution of the
thereby obtaining graph
5.2.4 (except that we do
solution).
G 20. y" + xy' +2.
G 21. (4-x2)y" +
G 22. y" + x²y = 0
G 23. (1-x) y" +=
5Charles Hermite (1822-1
algebraist. An inspiring teac
and the Sorbonne. He introdu
1873 that e is a transcendenta
equation with rational coeffic
matrices (see Serti
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