Math 314 Exam 1

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Math 314 Exam 1

W.T. Kiley 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 . Compare the result with J .

0 1

5. Show that between 2 zeros of J there is a zero for J .

0 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 ?