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Unformatted text preview: —.I_._J_... form. ‘ MATH2010/2100 Assignment 2
and Tutorial Set 2 l; p.150: Questions 2, 3, 4, 5, '6,
8. g . v __
7 K p.163: Questions Eli, 13*. Answer each starred problem as written and also sketch a‘phase
portrait in each case. For Q. 14 ﬁnd thegeneral solution in real Solutions to tEe two starred Eroblems to be
handed in at e en 0 your utorla or placed
in your assignment box on Level 3, Building
67, by ,5pm Monday, August 11.  NB: Use a cover sheet! (Download from the
Blackboard site.) ‘ lEG CHAP. 4 _ Systems of QDEs. Phase Plane. Qualitative Methods EXA MPLE 2 Free Motions of a Mass on a Spring '. To get a system, set y1 = Solution. Division by m gives y" = ~(k/m)y  (c/m)y
4.1). Then y; = y" — —(k/m)y1  (clm)y2. Hence 0 l
y’=[ 3y, det(A)\I)=\
I 2 c k
=A +—)\+'_—=0.
"l m — k/m ~ c/n
btain the following ' s and Tables 4.1 and 4.2 we 0 = k/m, A = (elm)2 — 4klm. From thi
ntial role. = clm,
the discriminant A plays an esse We see that p
the last three cases results. Note that in a stable and attractive spiral point. No damping. c = 0, p = , .
0, q > 0, A < 0,
d attractive node. Underdamping. 02 < 4mk, p <
Critical damping. c2 = 4mk, p < 0, q > 0, A = O, a stable an
Overdamping. c2 > 4mk, p < 0, q > O, A > riable) What happens to the
introduce r = —t al point. Then Determine the type
lution and sketch or graph some of the
15. (Types of critical points) Discus ﬁnd a real general so trajectories in the phase plane. (Show the details of your
(lO)—(l4) in Sec. 4.3 by applying the cr
4.1 and 4.2 in this section. work.)
1 yr 2 2Y2
16. (Perturbation of center) If a s
' its critical point, what happens if you replace the matrix
A by A = A + kl with any real number k as 0' I . . .
Q y1 = 2y1 + ya ‘ ' (representing measurement errors in the diagonal
__ entries)? 1e 4, Sec. 4.3, by ajk l .
that you get (a) a saddle point, (b) a stable and attractive
node, (c) a stable an al, (d) an unstable (1 attractive spir
spiral, (e) an unst able node.
18. CAS EXPERIMENT. Phase Portraits. Graph phase 5
portraits for the systems in Prob. 17 with. the values of
I) suggested in the answer. Try to illustrate how the phaSe
portrait changes “continuously” under a continuou. change of b. 19. WRITING
concepts are the following two—part report
applications in w s the critical points in and stability of the critic
iteria in Tables ystem has a center as ‘ Yl‘z z 8Y1 EXPERIMENT. Stability. Stability
basic in physics and engineering. Write a
of 3 pages each (A) on general hich stability plays a role (be 35
'al related to 5632) some or raarecrostes What kind of curVes are the trajectories of ODEs in the phase plane?
’ P
stability in th1 10. y" + 5 y = 0
examples; do not copy.
s of ilk to the critical point '
1,3, 5\ 11. y" — kzy = 0
12. y" + J—
20. (Stability chart) Loca
systems (10)—(l4) in Sec. 4.3 and of Probs.
this problem set on the stability chart. 13. (Damped oscillation) Solve y" + 4y’ + 5y = o. What
kind of curves do you get as trajectories? :L 22. y; I 3’2 olution. Determine the kind and stability of
(Show the details of your work.) 12. y'1=
7
3’2 I /='3y1 + 2y2 18. y'1 + 3)’2 ‘yl + 2y2 ‘23’1 — 3’2 Find a general solution. (Show the details.)
21. y'l = y1 + 2y2 + e” I ’ 3’2 y; = —y2 + l.5e'2t 12y1 + l y1+y2+sint 4)’1 + Y2 . (Mixing problem) Tank T1 in Fig. 99 contains initially
200 gal of water in which 160 1b of salt are dissolved.
Tank T2 contains initially 100 gal of pure water. Liquid
is pumped through the system as indicated, and the
mixtures are kept uniform by stirring. Find the amounts
of salt y1(t) and y2(t) in T1 and T2, respectively. Water.
10 gal/min 6 gal/min Mixture,
O gal/min Fig. 99. Tanks in Problem 26 27. (Critical point) What kind of critical point does y' = Ay
have if A has the eigenvalues 6 and 1? 28. (Network) Find the currents in Fig. 100, where
R1 = 0.5 0, R2 = 0.7 0., L1 = 0.4 H, L2 = 0.5 H,
E = 1 kV = 1000 V, and [1(0) = 0, [2(0) = O. Fig.100. Network in Problem 28 29. (Network) Find the currents in Fig. 101 when R = 10 0.,
L = 1.25 H, C = 0.002 F, and [1(0) = [2(0) = 3 A. Fig. 301. Network in Problem 29 W' 2 I (a j J” 2 [177w wwcf «'MM‘KJW, r“ “7% ﬁém “WM (74 MM—xﬂm we...)
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