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Exam 4

    Show all work: do not just give an answer.  If you use a theorem in the book, state which one you used.  Only dead sources are allowed.  If you use either Maple or Matlab, turn in the worksheets, but also write up by hand what you got for an answer and how.  Turning in just an uncommented worksheet will not be worth much since I may not understand what you did.  Be crystal clear even if it means writing more than you think is necessary.  Each part is worth 10 points for a total of 140.

    Turn in this sheet with your work to me in 321A McVey or in 325 McVey (CCS) no later than 8pm on Tuesday, December 14.  Earlier will be appreciated.  I will have 327 McVey open from 6-8pm on Tuesday, December 14 for anyone who wants to take the exam then.  There are a few computers hooked up to the Internet there, but none with Maple or Matlab on them.  Good luck.

1. Find the solution to the differential equations (solve for the constants, too)

(a) (t-1)y'' -3ty' + 4y = sint, y(-2) = 2, y'(-2) = 1
(b) y'' + 2y' + 1.25y = 0, y(0) = 3, y'(0) = 1
(c) 4y'' + 4y' + y = 0, y(0) = 1, y'(0) = 2
(d) y'' + 4y' +y = tet + 4, y(0) = 1, y'(0) = 1

2. Given (1-x)y'' + y = 0, 

(a) Find the solution by means of a power series about the point x0 = 0.
(b) Find the recurrence relation.
(c) Find the first four terms of two linearly independent solutions.

3. Use Maple to solve this problem (i.e., no Matlab).  

(a) Use dsolve with the series option to solve the Euler equation x2y'' + 3xy' + y = 0.
(b) Are the solutions singular at 0?
(c) What happens if you specify an initial condition at x = 0 in the dsolve command?

4. Find the Laplace Transform of sin(bx), where b is a real constant.

5. Use the Laplace Transform to

(a) Solve y'' - y' - 6y = 0, y(0) = 1, y'(0) = -1.
(b) Graph the solution.
(c) What happens at infinity?

 

 

Cheers,
Craig C. Douglas

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