Disc5 - Math 17C Kouba Discussion Sheet 5 1 The...

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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 atlmz-plane, 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 snag-coordinate 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—d-tl=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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