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But let's pretend we don't and try to un derstand why we would expect a solution, and why we would expect it to be unique, just based on the geometric meaning of this IVP: If we plo t the solution curve [(x(t),y(t)] in the plane, it passes through [1,0 ] at t=0, and then moves so that its tangent vector " }{TEXT 260 3 " [ x" }{TEXT 261 1 "'" }{TEXT 262 5 "(t),y" }{TEXT 263 1 "'" }{TEXT 264 4 "(t)]" }{TEXT -1 205 " agrees with the DE vector field at the point \+ [x(t),y(t)], namely with [y,-1.01x-.2y]. So if we create a picture o f the vector field we can visualize where the solution trajectory must start and then go:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "resta rt:with(DEtools):with(plots):with(linalg):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "fieldplot([y,-1.01*x-.2*y],x=-1..1,y=-1..1.);" }} {PARA 13 "" 1 "" {GLPLOT2D 319 289 289 {PLOTDATA 2 "6$-%'CURVESG6\\dl7 %7$$!3*=70()f7Zm*!#=$!3G!ocbxp0/\"!#<7$$!3#z[H,uGN.\"F-$!37(>LWC-Vf*F* 7$$!3MYTm8G:H5F-$!3\\2l/wjz`)*F*7%7$$!3``xv>537')F*$!3$evz590q.\"F-7$$ !3u4vMAel#G*F*$!3oTC?*e[*H'*F*7$$!3V$y,:(\\)*H#*F*$!3nRYv+u+u)*F*7%7$$ !3;&Q53W\\%fvF*$!3QJGg10WL5F-7$$!3QT,SVU-I#)F*$!3M(orR$\\fl'*F*7$$!3P. @O1=Wo\")F*$!3'HxiaU=U*)*F*7%7$$!3!o,j='y\"o]'F*$!3q1f7se()H5F-7$$!3,t FXkERxrF*$!3*=$4uy7C,(*F*7$$!3VCCAT')*o5(F*$!3;04<]%HW\"**F*7%7$$!3W[c \"HG'=aaF*$!3C#)*[wB6j-\"F-7$$!3m/a]&3hZ7'F*$!3Xw,^Bw)ot*F*7$$!3PWF3wa NXgF*$!3MP!z[ZSY$**F*7%7$$!3_z#oRqa:S%F*$!3yd?<.muA5F-7$$!3HO!el]H@2&F *$!35A%z#oR`s(*F*7$$!3([1V4J7Q)\\F*$!3kqre*\\^[&**F*7%7$$!3;64-DJ#*[LF *$!3LL^po>=>5F-7$$!3[o1hFz\\>SF*$!3mm'[IJ!=3)*F*7$$!3P&Q.e9pA#RF*$!3#G I&HCD1v**F*7%7$$!33VN2Y:H'H#F*$!3))3#=ULF*$!3)o:(e-IZz)*F*7$$!3OES_:G=*z\"F*$!3$o:rt X[:+\"F-7%7$$!3$G1)y\")QG5>!#>$!3ufVEl!)[35F-7$$!3@Jco2>.;')Fft$!3W,kN Z$>^\"**F*7$$!3!fYVQ]'RwtFft$!30q>%)f&pN+\"F-7%7$$\"3]@co2>.;')Fft$!3H NuyIM#\\+\"F-7$$\"3.^!)y\")QG5>Fft$!35Zc7#pl2&**F*7$$\"36HLbZ^.RKFft$! 3\\$y7Bm!f05F-7%7$$\"3lIfrpZB9>F*$!3$3^5jze8+\"F-7$$\"3Kth7n*fOC\"F*$! 3l\"*[*o.7k)**F*7$$\"3T7]*)zmW&Q\"F*$!3r'f$ykpM]%)*)pW#F *$!3$*4WDnGj45F-7%7$$\"3Pn1hFz\\>SF*$!38=mct_HU**F*7$$\"3054-DJ#*[LF*$ !33QLks/x05F-7$$\"3prV<5I`3NF*$!39B_spRl65F-7%7$$\"3=N!el]H@2&F*$!3ett zG*[m!**F*7$$\"3Ty#oRqa:S%F*$!3vi-72^L45F-7$$\"3>^SJvh2qXF*$!3eOg>s]n8 5F-7%7$$\"3a.a]&3hZ7'F*$!3\"z7GSe-5()*F*7$$\"3LZc\"HG'=aaF*$!3@(=(fT(* *G,\"F-7$$\"3!=t`/M>;j&F*$!3!)\\omuhp:5F-7%7$$\"3!>x_Wm#RxrF*$!3O$))e# RiNN)*F*7$$\"3q:I'='y\"o]'F*$!3m6T2wVY;5F-7$$\"3u5Mf0D;$p'F*$!3-jw8xsr <5F-7%7$$\"3FS,SVU-I#)F*$!3!)Q'*[%*)4(*z*F*7$$\"30%Q53W\\%fvF*$!37O5b5 !H+-\"F-7$$\"3!34L2n0Zv(F*$!3Cw%3'z$Q(>5F-7%7$$\"3j3vMAel#G*F*$!39$R?( \\N1k(*F*7$$\"3U_xv>537')F*$!3egz-XOfB5F-7$$\"3vpF(e$)[i\"))F*$!3Y*Gz? 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F*$\"3`pre*\\^[&**F*7%7$Fjz$\"3C#)*[wB6j-\"F-7$Fez$\"3Mv,^Bw)ot*F*7$$ \"3EVF3waNXgF*$\"3BO!z[ZSY$**F*7%7$Fj[l$\"3q1f7se()H5F-7$Fe[l$\"3yI4uy 7C,(*F*7$$\"3KBCAT')*o5(F*$\"3//4<]%HW\"**F*7%7$Fj\\l$\"3:JGg10WL5F-7$ Fe\\l$\"3B'orR$\\fl'*F*7$$\"3E-@O1=Wo\")F*$\"3'=xiaU=U*)*F*7%7$Fj]l$\" 3hb(z590q.\"F-7$Fe]l$\"3cSC?*e[*H'*F*7$$\"3K#y,:(\\)*H#*F*$\"3cQYv+u+u )*F*7%7$Fj^l$\"3G!ocbxp0/\"F-7$Fe^l$\"3+'>LWC-Vf*F*7$$\"37YTm8G:H5F-$ \"3Q1l/wjz`)*F*-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PAGEBK }{PARA 0 "" 0 " " {TEXT -1 308 " Now, pretend (?) you don't know how to find the \+ analytic solution to our page 1 initial value problem. We can use Eul er's method to get a numerical solution! In other words, approximate \+ using the tangent line approximation, and replace the tangent vector b y what the differential equation says it is:" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "matrix([[x(t+Delta*t)], [y(t+Delta*t)]]) = matrix([[x(t )], [y(t)]])+Delta*t*matrix([[diff(x(t),t)], [diff(y(t),t)]]);" "6#/-% 'matrixG6#7$7#-%\"xG6#,&%\"tG\"\"\"*&%&DeltaGF.F-F.F.7#-%\"yG6#,&F-F.* &F0F.F-F.F.,&-F%6#7$7#-F*6#F-7#-F36#F-F.*(F0F.F-F.-F%6#7$7#-%%diffG6$- F*6#F-F-7#-FG6$-F36#F-F-F.F." }{TEXT -1 0 "" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "matrix([[x(t+Delta*t)], [y(t+Delta*t)]]) = matrix([[x(t )], [y(t)]])+Delta*t*matrix([[y(t)], [-1.01*x(t)-.2*y(t)]]);" "6#/-%'m atrixG6#7$7#-%\"xG6#,&%\"tG\"\"\"*&%&DeltaGF.F-F.F.7#-%\"yG6#,&F-F.*&F 0F.F-F.F.,&-F%6#7$7#-F*6#F-7#-F36#F-F.*(F0F.F-F.-F%6#7$7#-F36#F-7#,&*& -%&FloatG6$\"$,\"!\"#F.-F*6#F-F.!\"\"*&-FL6$\"\"#FRF.-F36#F-F.FRF.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 439 "With the specified initial conditions we start at the po int [1,0], and then follow the tangent field around a curve which loo ks a lot like a spiral. If we want to get a numerical approximation, \+ we can just transpose Euler (or improved Euler or Runge-Kutta) for fir st order equations to first order systems . . . all the original justi fications make just as much sense in vector form!!! (Or we can use on e of Maple's numerical packages.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "f1:=(x,y,t)->y:\nf2:=(x,y,t)->-1.01*x-.2*y:\n #our \+ right hand side F(X,t)=[f1(X,t),f2(X,t)]\n #in our first order system of DE's dX/dt = F(t,X)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "n:=100:\n #number of time steps\nt0:=0: #initial time\ntn:=evalf (2*Pi):\n #final time choice\nh:=(tn-t0)/n:\n #time step\nxs:=vector (n+1):\nys:=vector(n+1): #vectors to hold Euler approximates\nxs[1]: =1:\nys[1]:=0: #initial conditions\n " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 10 "Euler loop" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 224 "#Euler loop:\nfor i from 1 to n do\n X:=xs [i]:\n Y:=ys[i]:\n t:=t0+(i-1)*h:\n k1:=f1(X,Y,t):\n k2:=f2(X,Y,t) :\n #k1 and k2 are velocity components\n xs[i+1]:=X+h*k1:\n ys[i+ 1]:=Y+h*k2:\n #update xs and ys with Euler!\nod:\n" }{TEXT -1 0 "" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 446 "tangefield:=fieldplot([y,-1.01*x-.2*y],x=-1..1.01,y= -1..1.):\nalpha:=arctan(.1):\nC:=1/cos(alpha):\nEulerapprox:=pointplot (\{seq([xs[i],ys[i]],i=1..n+1)\}):\nexactsol:=plot([C*exp(-.1*s)*cos(s -alpha),\nC*exp(-.1*s)*(-.1*cos(s-alpha)-sin(s-alpha)),\n s=0..2*Pi], color=black): #This is the actual\n #solution to the IVP, more compl icated than\n #the Friday formula, but correct, I hope\ndisplay(\{tan gefield,Eulerapprox,exactsol\},title=\"go with the flow\");" }}{PARA 13 "" 1 "" {GLPLOT2D 254 241 241 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$\"2M$ e)))*********!#<$!3P\"Re\\MaO<\"!#H7$$\"3E9V%e![G1**!#=$!3yzcjrW/e5w'*F1$!3]OZ-QD#R\\#F17$$\"3=__LOiZg#*F1$!3OGE&e^rTp$F17$$ \"3O(ofuD!3'o)F1$!3t'yD7p,O![F17$$\"36Bo=rDTtzF1$!3'4c?W]-@z&F17$$\"3F fut&f'R,sF1$!3c0Uu*\\QEf'F17$$\"36<[(\\9u]I'F1$!3_.*eSunMH(F17$$\"3y%f `F-0EH&F1$!3HUG)RRJD(yF17$$\"3gf>(H5Mn@%F1$!3%=E'3I'\\gH)F17$$\"3EyTXT `TiIF1$!33qai')R,o&)F17$$\"3)>$=$!3Q[I%\\l18j)F17$$!3#ovmf%zYMJFco$!3A`EWp1DN%)F17$$!3obV&f'*o$* R\"F1$!3PRIC'**eT5)F17$$!3[o;\"fZG3M#F1$!359\\G<&\\/p(F17$$!3o2)4VTnuQ $F1$!3wUy;YIXqqF17$$!3s9ee#))*3*>%F1$!3bh#\\Hp-yW'F17$$!3UC?w=mgU]F1$! 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$!+CW(H,'Fi[el$\"+2xu,aFi[el7$$\"+*p]CN%Fi[el$\"+?\\')H\\Fi[el7$$\"+ZK ?iYFi[el$\"+Urq\"f%Fi[el7$$!+1EdtcFi[el$\"+Kaw\")QFe]el7$$!+7L\"Q2)Fi[el$\"+EA>v 7Fe]el7$$!+Vkev9Fi[el$\"+aMsNtFi[el7$$!+X$pY,\"Fi[el$\"+.5=PtFi[el-%+A XESLABELSG6%Q!6\"Fa[gl-%%FONTG6#%(DEFAULTG-%&TITLEG6#Q1go~with~the~flo wFb[gl-%%VIEWG6$Ff[glFf[gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 385 " Any first order system is analogous to the above example, and that's why we expect existence and uniqueness in general. As wit h first order DE's, we'll get better approximate solutions using more \+ sophisticated algorithms, like improved Euler and Runge Kutta. For ex ample, here's some improved Euler code, first for the first order scal ar ODE, then converted to the system above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 28 "Original improved Euler loop" }{TEXT -1 23 ", stolen from the file " }{TEXT 265 14 "numerical1.mws" }{TEXT -1 24 ", in our home directory:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "for i from 1 to n do\n k1:=f(x,y): #left -hand slope\n k2:=f(x+h,y+h*k1): #approximation to right-hand slo pe\n k:= (k1+k2)/2: #approximation to average slope\n y:= \+ y+h*k: #improved Euler update\n x:= x+h: #up date x\n print(x,y,exp(x));\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 233 " We now vectorize everything, and store the [x,y] values in a vector so we can make a picture like we did for unimproved Euler. (Before exec uting the code below I re-entered all the initialization definitions f rom the previous page.)" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 272 19 "Impr oved Euler loop" }{TEXT -1 54 " for first order system of two differen tial equations:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "#Improve d Euler loop:\nfor i from 1 to n do\n X:=xs[i]:\n Y:=ys[i]:\n t:=t0 +(i-1)*h:\n k11:=f1(X,Y,t):\n k12:=f2(X,Y,t): \n #k11 and k12 are left-hand tangent estimates\n k21:=f1(X+h*k11,Y+h*k12,t):\n k22:=f2 (X+h*k11,Y+h*k12,t):\n #k21 and k22 are right-hand tangent estimate s\n xs[i+1]:=X+h*(k11+k21)/2:\n ys[i+1]:=Y+h*(k12+k22)/2:\n #upda te xs and ys with improved Euler!\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 473 "tangefield:=fieldplot([y,-1.01*x-.2*y],x=-1..1.01,y= -1..1.):\nalpha:=arctan(.1):\nC:=1/cos(alpha):\nEulerapprox:=pointplot (\{seq([xs[i],ys[i]],i=1..n+1)\}):\nexactsol:=plot([C*exp(-.1*s)*cos(s -alpha),\nC*exp(-.1*s)*(-.1*cos(s-alpha)-sin(s-alpha)),\n s=0..2*Pi], color=black): #This is the actual\n #solution to the IVP, more compl icated than\n #the Friday formula, but correct, I hope\ndisplay(\{tan gefield,Eulerapprox,exactsol\},title=\"go with the flow better with Im proved Euler\");" }}{PARA 13 "" 1 "" {GLPLOT2D 288 288 288 {PLOTDATA 2 "6(-%'CURVESG6$7S7$$\"2M$e)))*********!#<$!3P\"Re\\MaO<\"!#H7$$\"3E9 V%e![G1**!#=$!3yzcjrW/e5w'*F1$!3]OZ-QD#R\\#F17$$\"3=__ LOiZg#*F1$!3OGE&e^rTp$F17$$\"3O(ofuD!3'o)F1$!3t'yD7p,O![F17$$\"36Bo=rD TtzF1$!3'4c?W]-@z&F17$$\"3Ffut&f'R,sF1$!3c0Uu*\\QEf'F17$$\"36<[(\\9u]I 'F1$!3_.*eSunMH(F17$$\"3y%f`F-0EH&F1$!3HUG)RRJD(yF17$$\"3gf>(H5Mn@%F1$ !3%=E'3I'\\gH)F17$$\"3EyTXT`TiIF1$!33qai')R,o&)F17$$\"3)>$=$!3Q[I%\\l18j)F17$$!3#ovmf%zYMJFco$!3A `EWp1DN%)F17$$!3obV&f'*o$*R\"F1$!3PRIC'**eT5)F17$$!3[o;\"fZG3M#F1$!359 \\G<&\\/p(F17$$!3o2)4VTnuQ$F1$!3wUy;YIXqqF17$$!3s9ee#))*3*>%F1$!3bh#\\ Hp-yW'F17$$!3UC?w=mgU]F1$!3Q)R$o.<`EcF17$$!3#oFX%3(*z)o&F1$!3#)Qt(R!e! p#[F17$$!3IO$f))))G*ziF1$!3)e>%*oF17$$!35A*G%e`\"[B(F1$!3'yf-H;%Q<5 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