Differential Equations, mat 3613
Final, December 15, 1995
Instructor: D. Gokhman
Name: Pseudonym:
Show all work. Answers alone are not sufficient. Box the answers.
1. (80 pts.) Solve the following initial value problems and describe the behaviour of
each solution for large x.
(a) y00 +4y0
−5y=0,y(0) = 1, y0(0)=0
(b) y00 +4y0+5y=0,y(0)=1,y0(0) = 0
(c) y00 +4y0+4y=0,y(−1) = 2, y0(−1)=1
(d) y000 +2y00
−5y0
−6y=0,y(0) = 0, y0(0) = 0, y00 (0)=1
[Hint: Find one characteristic root by trial and error.]
2. (40 pts.) Find the general solution for each of the following equations:
(a) y00 +y=x(1 + sin x)
(b) y00 +2y0+y=e−xlog x
3. (30 pts.) Find the terms up to and including x5of the power series solution for
the initial value problem (x−1)y00
−xy0+y=0,y(0) = −2, y0(0) = 6
Extra credit: find the general form of the series and compute its radius of conver-
gence.
1a 1b 1c 1d 2a 2b 3 total (150)
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