newproject2posted.pdf - Project 2 Math 306 — DUE APRIL 10...

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Unformatted text preview: Project 2 - Math 306 — DUE APRIL 10 Spring 2018 Prof. Dimoek Consider the following differential equation J 3 .17”+ 3’— ~ :1: + m— z 0 5 6 1. Write the equation as a. first order system :3’ = f (:B,y),y’ = g(.1:,y). Find all the equilibrium points. (An equilibrium point hey) is a point such that the vector field vanishes: f(x,y)=0, g(x,y)=0). 2. Using Maple plot the direction field. 3. Using Maple and Euler’s method plot solution curves (560?), y(t)) with various choices of initial conditions. (Plot enough curves and use a small enough step size to get a good idea of the phase portrait. You need not display tables of the data points) 4. Plot the graphs of 1305) and y(t) for at least one choice of initial condi— tions. Notes: 1. On the next page we include some samples which you can use as a guide. The sample is for the van der Pol equation which we did in class. The step size in the samples is not very small, you will want to take something smaller. The samples are presented first in 2-d input and then in 1—d input (Maple input).You can work in either. 2. Hint: to start a. new line without executing the commands, press shift key with return. 3. Recitation sections will meet in the computer lab Baldy 8B the week of April 2-6. restart : with (DEtools) : I plot] == dfieldplafH-ST x(r) =y(t), % y(f) = —x(r) + (I —x(f)2) y(f)], [Jay], 1': 0..1,x= —1-.6..1.6,y= -1.6..1.6) : with (plots) :display(plorl ) (((‘////// (((////// (//////// ~*\V\\\\C\\\\V\ \x\\\“’\\\\\ tend h J'ca’ata0 == x :ydcu’aD == y : forkfrom 0 to n -— 1 do flc==y; gk== —x+ (1 —x2)y; t==f+h; x==x+hfk; ymy+hfl; tdatak + 1 == 1' xdat-ak + 1 == x; ydatak + l == y end do : forifrom 0 to 11 do restart : 11 == 0.25 :tend == 2.0 : n := [ l: r == 0. :2: == 1.0 :y := 1.0 : tdatao := t: print“ "%d %8.2f %12.5f %12.5f\n", 1', tdatai, xdataf, ydataf) end do : 0 0.00 1.00000 1.00000 1 0.25 1.25000 0.75000 2 0.50 1.43750 0.33203 3 0.75 1.52051 —0.11586 4 1.00 1.49154 —0.45799 5 1.25 1.37704 —0.69065 6 1.50 1.20438 —0.88016 7 1.75 0.98434 —1.08212 8 2.00 0.71381 —l.33661 wdamlisf == [seq( [xdafap ydafaf], 1': 0 .17) 1 : plothydatalist, color = 'blue') 1 0.5 -0.5 txdamlist := [seq( [tdatap xdafai], i= 0 ..n)] : plotdeatalisf, color = 'green') 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.5 1.5 > restart. with(DEtools): Ia plot1:= dfieldplot( [diff(x(t), t)=y(t), diff(y(t), 6t)=-x(t) + (1-x (t)‘2)*y(t)], [x y], t=0. .1, x=—1. 6. .1. 6 ,y=-1. 6. .1. h) with(p10ts): display(plot1); \fl2//////////2a\\\\\ \12/////}-///2»\\\\\ NA///////////2u\\\\\ »////////{///»~\\\\\ »//////;/////A\\\\\\ /////////////H\\\\\\ /////////g//»\\\\\\\ ////////P7//fi\\\\\\\ IIIl/l/////A\\\\\\\\ - m 1 - 0. i .5 1 .5 «\«\\\\\~ //§//Jll l x\\\\\\~fi{////////// \'\'\\\\\-r-*/‘/////////// \'\\\\‘\*-—(/////////// \\\\\\e/4{/////////r \\\\‘\“--=-’///////////z~r—- \\\\\««//////////zex \\\\\hng{///////x~\ \\\\\LL/ 7///////(h\ > restart. h:= 0.25 tend:= 2.0: n: a ceil(tend/h): t:= 0.0: x: = 1.0: y: = 1.0: tdata[0]. -= xdata[0]. = ¥ggta[g;om§ 0 to n-1 do fk :- y. gk := — x+ (1-x‘2)*y. t := t+h: x := x + h*£k: y := y + h*gk: tdata[k+1]:= t: xdata[k+1]:= x: ydata[k+1]:= y: ed: for i from 0 to n do no printf( 6 od: 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 xydatalist := V mummprI—Io %3.2£ %12.5£ %12.5f\n",i,tdata[i],xdata[i],ydata[i]) 1.00000 1.00000 1.25000 0.75000 1.43750 0.33203 1.52051 —0.11586 1.49154 —0.45799 1.37704 —0.69065 1.20438 —0.88016 0.98434 —1.08212 0.71381 —l.33661 [seq ( Exdata[i], ydata{i1], i=0..n )]: plot( xydatalist, color='b1ue' ); 1 (LS -(LS > txdatalist := [seq ( [tdata[i], xdata[i]], i=0..n )]: plot( txdatalist, color='green ); 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.5 1.5 ...
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