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
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 yı
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.
Xo = 0
1.
y" - y = 0,
2.
y" + 3y = 0,
3.
y" - xy' - y = 0,
xo = 0
xo = 1
4. y"-xy' - y = 0,
5. y"+k²x²y = 0,
6. (1-x)y"+y=0,
7. y" + xy' + 2y = 0,
8. xy"+y' + xy = 0,
xo = 0, k a constant
xo = 0
xo = 0
xo = 1
9.
(3-x²)y" - 3xy' - y = 0, Xo = 0
xo=0
10. 2y" + xy' + 3y = 0,
11. 2y" + (x + 1) y' + 3y = 0, xo = 2
xo = 0
In each of Problems 12 through 14:
a. Find the first five nonzero terms in the solution of the given
initial-value problem.
ons
Gb. Plot the four-term and the five-term approximations to the
solution on the same axes.
mount
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)² y' + (x² - 1) y = 0
17. Show directly, using the ratio test, that the two series solutions
of Airy's equation about x = 0 converge for all x; see equation (20)
of the text.
in powers of x - 1.
b. Show that you obtain the same result by assuming that y
has a Taylor series in powers of x - 1 and also expressing the
coefficient x² - 1 in powers of x - 1.
16. Prove equation (10).
18. The Hermite Equation. The equation
y" - 2xy' + Ay = 0, -∞0 < x < ∞,
where A is a constant, is known as the Hermites equation. It is an
important equation in mathematical physics.
OST 19. Consider the initial-value problem y'=√1-y2, y(0) = 0.
a. Show that y = sinx is the solution of this initial-value
problem.
b. Look for a solution of the initial-value problem in the form of
a power series about x = 0. Find the coefficients up to the term
in x3 in this series.
a. Find the first four nonzero terms in each of two solutions
about x = 0 and show that they form a fundamental set of
solutions.
b. Observe that if X is a nonnegative even integer, then one
or the other of the series solutions terminates and becomes a
polynomial. Find the polynomial solutions for λ = 0, 2, 4, 6,
8, and 10. Note that each polynomial is determined only up to a
multiplicative constant.
c. The Hermite polynomial H₂(x) is defined as the polynomial
solution of the Hermite equation with λ = 2n for which the
coefficient of x" is 2". Find Ho(x), H₁(x), ..
..., H5(x).
REIS
In each of Problems 20 through 23, plot several partial sums in a
series solution of the given initial-value problem about x = 0,
thereby obtaining graphs analogous to those in Figures 5.2.1 through
5.2.4 (except that we do not know an explicit formula for the actual
solution).
y"+xy' + 2y = 0, y(0) = 0, y'(0) = 1; see Problem 7
(4-x2) y" + 2y = 0, y(0) = 0, y'(0) =
G 22.
y" + x²y = 0, y(0) = 1, y'(0) = 0;
y(0) = 1,
y'(0) = 0; see Problem 5
y(0) = 0, y'(0) = 1
G 23.
(1-x) y" + xy' - 2y = 0,
siem nessWszitemsile.na att 02
G 20.
G 21.
no
Charles Hermite (1822-1901) was an influential French analyst and
algebraist. An inspiring teacher, he was professor at the École Polytechnique
and the Sorbonne. He introduced the Hermite functions in 1864 and showed in
1873 that e is a transcendental number (that is, e is not a root of any polynomial
equation with rational coefficients). His name is also associated with Hermitian
matrices (see Section 7.3), some of whose properties he discovered.
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