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Unformatted text preview: Math 17C
Kouba
Discussion Sheet 5 1.) The position (:r1,:r2) Of a particle at time t is given parametrically by each of the
following. Eliminate t and write each as an equation in only $1 and 332 . Then sketch the
graph of the path in the atlmzplane, indicating the direction of motion of the particle. a) ' :r2=2t—5, for —oo<t<oo. 1:1 = lnt
r2 = (lnt)3 — 2(lnt)2, for t > 0. 1
1'2: 4—t, forOStS4. :vgzsint—B, for0_<_tg27r. :v —t2
c.) 1— _
{1722256—2134, ior—oo<t<oo.. f.) (Challenging) 5’51 : t2 — 2t
1:2:t2+t, for_oo<t<oo. 2.) Use the parametric graphing function on a graphingcalculator to plot the following
path. Then ﬁnd a unit vector tangent to the path, the direction of motion, and the speed
of motion when t = 71' b.) t: 77r/2. 1171 = tcost
1'2 2 tsint, for 0 S t g 471'. Write the following system of differential equations in matrix (vector) form. ([271 _ 7
ddt _ $2
CL‘
—dt—2 = 311:1 — £132 Write the following system of differential equations in parametric form. 2 —1
(4 3)X ‘ :L'1 = 5 cos 3t . . .
’. h " __ ,  ,  c _
a ) f3 ow that {m2 : 4COS 3t +381n3t solves the followmg system of diffeiential equa XI tions : da: 1 — = 4: — 5.’ 1d t E 1 12 £5? = 5151 —' 4CE2
6.) Show that X = et + tet solves the following system of differential equa
. _ , _ 2 1
tions. X — <_1 0 X 7.) (Creating a direction ﬁeld) Consider the following system of differential equations. For
each of the following pairs of points (5131, $2) set up a table to indicate the slope, direction
vector, and speed at that point. On an snagcoordinate system plot the direction vector
at each point and indicate the relative length (speed) of each vector. Use the following
points : (1,1), (1,2), (1,0), (1,1), (1,—2), (0,0), (0,1,), (0,2),(0, ~1), (0,2), '(1,1), (—1,2),
(‘19)) (‘1v‘1)7 (‘1a'2)a (37'3)7 (42) (£331 ‘dd—t:~2$1+332
$21171—2132 8.) Find the general solution to each of the following systems of differential equations.
Write your answer in matrix (vector) form. a.) X’: (“‘21 (1)>X b.) X’= ‘11)X —2 1 —3 3/4
x _ I _
c.)X —<1 _2)X d.)X —(_5 1>X
9.) Solve the following system of differential equations with initial conditions. Write your
answer in matrix (vector) form and parametric form. drc
d—dtl=r1+2sc2 , 731(0)=5
73:22— : 41:1 + 3.132 , "62(0) 2 —2 10.) The point (0,0) is an equilibrium for each of the systems in problem 8.) For each system determine if (0,0) is an unstable or stable equilibrium. Then categorize (0,0) as a
saddle, sink, or source. +++++++++++++++++++++++++++++++++ 7"If you judge people, you have no time to love them.”  Mother Teresa ...
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 Summer '09
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