Math 675
Discussion Questions
Chapter #6, Part 1
Reading: Chapter 6, Section 6.4
66. Find the leading behavior of
/2
ex tan t dt
0
as x .
67. Find the leading behavior of
2
ex sinh
t
dt
0
as x .
68. Find the leading behavior of
/2
(t + 2)ex cos t dt
/2
Math 675
Discussion Questions
Chapter #7, Part 1
Reading: Chapter 7, Sections 7.1, 7.2
73. Find a perturbation series solution for each of the roots of x3 + x2 x = 0 with terms up to and
including O( 3 ). Compare your results to numerical values from a CA
Math 675
Discussion Questions
Chapter #6, Part 1
Reading: Chapter 6, Sections 6.1 - 6.3
61. Consider the integral
1
I (x) =
cos(xt) dt
x
(a) This is suciently simple that the integral can be evaluated directly. Do so, and express
the result as a series in
Math 675
Discussion Questions
Chapter #3, Part 4
Reading: Chapter 3, Sections 3.5 - 3.8
54. Consider the parabolic cylinder equation
y + ( +
1
2
1 x2 )y = 0.
4
(See Example 4, pp. 9698.)
(a) Verify that y has an irregular singular point at innity.
2
(b)
Math 674
Discussion Questions
Chapter #3, Part 1
Reading: Chapter 3, Sections 3.1-3.3.
23. (1.7 ). Consider the equation
x2 y + 3xy + 2y = x4 y 3 .
(a) Let = ax and = a1 y , and show that the equation is unchanged.
(b) Write y = u/x and reduce the equatio
Math 674
Discussion Questions
Chapter #3, Part 2
Reading: Chapter 3, Sections 3.4
35. Prove each of the following:
1/x as x 0+.
(a) x
(b) 1/x
(c) x
2
x as x .
25 as x 0.
36. Prove or disprove: If f (x)
x x0 .
g (x) as x x0 and g (x)
h(x) as x x0 , then f
Math 675
Discussion Questions
Chapter #3, Part 3
Reading: Chapter 3, Sections 3.5
50. Suppose that f (x)
n=0
an xn . Is it the case that
n=0
an xn converges?
51. Suppose that
(a) f (x)
(b) g (x) =
n
n=0 an x
n
n=0 an x
and
with radius of convergence R >
Math 675
Discussion Questions
Chapter #9, Part 1
Reading: Sections 9.1-9.3.
76. Consider the problem
y + 2y + 2y = 0
y (0) = 0
y (1) = 1
(a) Find the exact solution. Use a CAS or otherwise to graph the solution for = 1, 1/10, 1/100.
What do you notice abo
Math 675
Discussion Questions
Chapter #9, Part 2
Reading: Sections 9.4
77. Consider the problem
y x2 y y = 0
y (0) = 1
y (1) = 1
(a) Look for an outer solution in the form
y (x) y0 (x) + y1 (x) + . . .
as
0+. Find the equations for y0 , y1 , . . . .
(b)