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Unformatted text preview: University of Waterloo Department of Applied Mathematics AMATH 351 Spring 2002
Midterm Test  Questions Instructor: J. Wainwright Time: 1% hours [15] 1. (a) Find the general solution of the DE d2y dy 2a
m‘ a” by making the change of variable 2 = e y=0. km (b) Verify that the method of Frobenius can be applied to solve the DE
2103/" + (3 —— $)y' — y = 0. Find the indicial equation and the general form of the recurrence relation. [16] 2. Consider the DE (i) Calculate the matrix exponential 6“. State any results you use. (ii) What is the long term behaviour of the solution with initial condition
x0 = (a, b, c, 0)? What happens if this initial condition is perturbed slightly to
x0 = (a,b,c,10‘3)? (iii) Show that the subset of R4 deﬁned by
$21) + x3 = 023%, where C is a positive constant is an invariant set of the DE. (iv) Give a qualitative sketch of the orbits of the DE in the invariant set x3 = 0 = x4,
and in the invariant set :04 = 0. AMATH 351  Midterm Examination Spring Term 2002 I 4~ , ,V [14] 3. Consider the DE
57 I y” + x ‘4 2(b—1)y = 0 where b is a positive constant. Use the properties of Bessel functions to answer the
following questions: W Show that all solutions are bounded as 17 —> 0+. (ii) It is desired that all solutions tend to 0 as a: —+ +oo. Find a restriction on b that
will guarantee this result. (iii) Show that all solutions have inﬁnitely many zeros. Describe the spacing of the
zeros as at —> oo. [15] 4. (a) Use the deﬁnition of the matrix exponential to derive an expression for
’ ' d ——(e‘tA), where A is a constant n X 17. matrix. dt
(ii) Show that the DE x’ = Ax + f(t), where A is a constant n x 72 matrix and x 6 IR", can be written in the equivalent form
[€wa = e—tAf(t). Hence show that the unique solution of (*) with initial condition x(0) = x0
is t
X(t) = emxo +/ €(t_T)Af(T)dT.
0
(b) Prove that any non—trivial solution of the DE
y”— (1+ e”)?! = 0 has at most one zero. State the theorem that you use. Bonus question Apply any methods with which you are familar to obtain information, either exact,
qualitative or heuristic, about the solutions of the DE y”7+ 2233/ + 2y = 0. Page 2 of 2 ...
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This note was uploaded on 09/20/2010 for the course AMATH 351 taught by Professor Sivabalsivaloganathan during the Spring '08 term at Waterloo.
 Spring '08
 SivabalSivaloganathan

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