18.305 FALL 2005 --- PROBLEM SET #1
POSTED:          Mon. 09-19-2005
DUE DATE:        Fri. 09-30-2005at NOON.
Turn in place:   Math. Undergrad. Office, room 2-108

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The problems below are from the book by Bender and Orszag, modified as
indicated.
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REGULAR PROBLEMS:
(3.4)  parts: (d), (e), and (f).
       WEIGHT= 30 points (10 points each part).
(3.6)  part (a). Note: write ALL the coefficients in the Taylor
       expansion explicitly.
       WEIGHT= 10 points.
(3.22) Ignore the part about evaluating the solution at y(1).
       Ignore the part about finding the exact solution in terms of
          Bessel functions.
       Do this:
       (a) Produce a solution in terms of a Taylor series at infinity.
           Namely, a solution of the form
           y = Sum_{n=0}^{n=inf} a_n x^{-n}
           for some coefficients a_n [compute them explicitly].
           There is only ONE linearly independent solution of this
           form.
       (b) A second linearly independent solution of the equation has
           a leading order behavior y(x) ~ x. What is the next largest
           term in an expansion for x large and positive?
       (c) Place your answers to parts (a) and (b) in terms of the
           classification of singular points.
       WEIGHT= 30 points (10 points each part).
(3.32) This problem shows that dominant balance in the usual way does
       not always work. EXPLAIN why not.
       However: a different dominant balance in the equation for S
       that results from the substitution y = e^S gives an appropriate
       expansion for the solutions. SHOW this.
       WEIGHT= 30 points.
       This problem is non-trivial. You get 10 points for shooting in
       the right direction, and hitting rather close to the target.
       From then on it gets tougher. The full 30 points are for a
       complete answer, including the form of the correct expansion
       near the singular point, and a justification that it is indeed
       a consistent expansion [with each term of smaller order than
       the prior one].
       Consider anything about 10 points as "xtra" credit.  
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SPECIAL PROBLEMS:
(3.1)  Hint: think of what properties you need of the transformation
       sending infinity to zero for the definition to make sense.
       WEIGHT= 10 points.
(3.33) part a.
       You should be able to find 3 linearly independent solutions.
       Hint: when you substitute y = e^S in the equation, and then
       find an expansion for S, it is IMPORTANT to not just compute
       the first term in the expansion for S. You must compute ALL
       the terms in S that are not small as x --> 0+. Else you will
       not get the complete correct behavior for y as x --> 0+.
       WEIGHT= 10 points.
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