Download Math 314 Exam 1

Math 314
W.T. Kiley
Exam 1
February 13-20, 2001
This part of the exam is take home open book.
You may not discuss it with anyone.
It is due at 10:30 a.m. on February 20.
Clearly show your work and your answers.
2
1. Find a basis for the solutions of x y" -3xy' + 4(x + 1)y = 0 around x = 0.
Find a general term for one member or your basis and the first 3 terms
of the other.
2. Do 2 of the special cases in 4.4 # 16c.
3. 4.4 #20.
4. 4.5 # 8
5. 4.6 # 12, the part about approximating the zeros.
6. Find J (2) by using the recurrence relations in the text and A1 on page A85.
5
MAth 314
February 15, 2001
This part of the exam is closed book. You may use one page of notes.
2
2
2
a) x y" + xy' - y = 0; b) y" + xy' - x y = 0; c) x y" + (3x - 1)y' + y = 0.
1. One of these has an analytic solution satisfying y(0) = 1, y'(0) = 2.
Find the terms of the series through x to the 4'th.
2. One of these can be solved near 0 by the Frobenius method.
Use the method to find a basis for the solution space.
3. Use the method for the remaining equation. Find the radius of convergence.
4. Differentiate the power series of J .
0
5. Show that between 2 zeros of J
0
Compare the result with J .
1
there is a zero for J .
1
2
2
6. Use the change of variable z = 2x to solve x y" + xy' + (4x - 9)y = 0
in terms of Bessel functions.
x
7. Assume e
and sin x are a basis for the solution space of an equation.
2
2
a) Can 2 + 3x + 4x be the first terms of a solution? b) Can 2 + 3x + x ?