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Unformatted text preview: University of Waterloo
Faculty of Mathematics AMATH 351 — Spring 2006
Final Examination Instructor: D. Siegel Date: August 3, 2006
Aids: Formula Sheet & Calculator Time: 9:00 to 11:30 am. Instructions: Label two examination booklets 1 and 2. Put your name and ID number
on each. Put the answers to questions 1 through 4 in booklet 1 and the answers to questions 5 through 8 in booklet 2. 1 1. [10] Consider the second order linear DE y" + 4—3353; = 0, for x > O. (a) Verify that y1(:r) = ﬂ is a solution. (b) Use reduction of order to ﬁnd a second independent solution y2(a:).
)
) (c Give the general solution. 1 (d Show that any nontrivial solution to y” + q(a:)y = 0, Where q(:c) < E for a: > O, has at most one zero for a: > 0. 2. 15 Consider the problem 3:23;” + my' + 5y = 0, for a: > 0, lim y a: = 1.
4 m—>O+ (a) Show that the indicial equation is r2 = 0. 00
(b) Find the ﬁrst three nonzero terms of the Frobenius series solution y(m) = Z anm"
n=0
by ﬁnding a recurrence relation for the coefﬁcients. (0) Find the solution y(:r) in terms of Bessel functions. (d) Give the asymptotics of the solution y(:c) as a: —> 00 and sketch the graph of y(a:) —11 0 3. [13] Consider the vector DE x’ = Ax Where A = —1 —l O
0 0 —l (a) Find the fundamental matrix 6’“. (b) Find the solution x(t) with x(0) = a.
(c) Show that the aslxg—plane is an invariant set and sketch the orbits in the 33le—
plane. (d) Show that the cone x? + 93% = 03:3, 0 is a positive constant, is an invariant set
and use this to sketch the orbits in R3. 4. [9] Consider the time—invariant system x’ = Ax+f (t), x(0) = x0, where A = < 32 32 ) . (a) Show that A is a stable matrix. 0 t—>oo Jr (b) For f(t) = < 1 > , ﬁnd lim x(t). Does the limit depend onf x0? (c) For f (t) = ( €1_t > , ﬁnd lim x(t). How is this related to part (b)? t—+oo Change to Booklet 2 5. [10] Consider a driven oscillator modeled by y” + y = f(t), y(0) = 0, y’(0) = 0, t 2 0. (a) Use Laplace transforms to ﬁnd an integral formula for the solution. (b) Find y(t) if f(t) = 1 for 7r 3 t S 277, f (t) = 0 otherwise. Hint: consider the
intervals 15 < 7r, 7r 3 t S 27r and t > 277. (c) Sketch the solution found in part (b). 6. [16] A periodic impulsive force is applied to a damped mass—spring system. The motion
00 is governed by y” + 33/ + 2y = p266 — n), y(0) = 0, y’(0) = 0 and p > 0.
n=0
(a) Find the solution y(t) by using Laplace transforms. 4k (b) Find the periodic steady—state yper(t) so that tlim [y(t) —yper(t)] = 0. Hint: consider
—>00
the interval m < t < m + 1. + (0) Sketch yper(t). 7. [13] A nonlinear oscillator is modeled by y” + y + €(y + y2) = 0, y(0) = b, y’(0) = 0,
with0<b<1, 0<e<<1. (a) Use the LindstedtPoincaré method to ﬁnd an approximate solution to ﬁrst order
in 6. Give the approximation to the period. (b) What is the disadvantage in using a regular perturbation series? 8. [8] (a) Prove the result that if g(t) is continuous on the interval [(1, b], g(t) Z 0 and
t
g(t) S L/ g(s) d5 for t 6 [a, b], L > 07 then g(t) = 0 on [a, b]. Hint: consider h(t) = L/tg(s) ds. (b) State the important result that is proved using the result in part (a). ...
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