John Travis
Mississippi College
First we consider Euler's Method for Systems. This is the kind of method one can do relatively easy by hand and which is good for demonstration purposes. However, practically one would want to utilize numerical methods which yield greater accuracy. The cost is more calculations...not easily done by hand but no problem for a computer.
# Euler's method for systems
# Code below written perhaps inefficiently but to show
# how the solution points are generated
t,x,y = var('t,x,y')
numpts = 5
# de1(x,y) = y+y^2
# de2(x,y) = -x+y/5-x*y+6*y^2/5
# de1(x,y) = y
# de2(x,y) = -2*x-y/2
de1 = 2*x + y
de2 = -1*x+y
delt = 1/4
old_x = 1/2
old_y = 1/2
soln_x = [old_x]
soln_y = [old_y]
for t in range(1,numpts):
new_x = old_x+delt*de1(x=old_x,y=old_y)
new_y = old_y+delt*de2(x=old_x,y=old_y)
soln_x.append(new_x)
soln_y.append(new_y)
old_x = new_x
old_y = new_y
m = len(soln_x)
pts = []
for k in range(m):
point = [soln_x[k],soln_y[k]]
print point
pts.append(point)
# G = points(pts)+plot_vector_field((de1(x,y), de2(x,y)), (x,-1/2,2), (y,-1,1))
G = points(pts)
G.show()
[1/2, 1/2] [7/8, 1/2] [23/16, 13/32] [289/128, 19/128] [1753/512, -97/256] 
[1/2, 1/2] [7/8, 1/2] [23/16, 13/32] [289/128, 19/128] [1753/512, -97/256]
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# Euler's method for systems
# Code below written perhaps inefficiently but to show
# how the solution points are generated
var('t,x,y')
de1(x,y) = y+y^2
de2(x,y) = -x+y/5-x*y+6*y^2/5
# from the inside spiraling out
delt = 0.05
old_x = 1
old_y = 1
soln_x = [old_x]
soln_y = [old_y]
for t in range(1,1000):
new_x = old_x+delt*de1(old_x,old_y)
new_y = old_y+delt*de2(old_x,old_y)
soln_x.append(new_x)
soln_y.append(new_y)
old_x = new_x
old_y = new_y
m = len(soln_x)
pts1 = []
for k in range(m):
point = [soln_x[k],soln_y[k]]
print point
pts1.append(point)
G = points(pts1,color='red')
# And from the outside spiraling in
delt = 0.05
old_x = .01
old_y = .01
soln_x = [old_x]
soln_y = [old_y]
for t in range(1,1000):
new_x = old_x+delt*de1(old_x,old_y)
new_y = old_y+delt*de2(old_x,old_y)
soln_x.append(new_x)
soln_y.append(new_y)
old_x = new_x
old_y = new_y
m = len(soln_x)
pts1 = []
for k in range(m):
point = [soln_x[k],soln_y[k]]
print point
pts1.append(point)
G += points(pts1,color='blue')
G +=plot_vector_field((de1(x,y), de2(x,y)), (x,-1/2,2), (y,-1,1), color='yellow')
G.show()
WARNING: Output truncated!full_output.txt [1, 1] [1.10000000000000, 0.970000000000000] [1.19554500000000, 0.927804000000000] [1.28497621312080, 0.873492434085960] [1.36680028644537, 0.807637039708032] [1.43979601782619, 0.731316124196504] [1.50310298771150, 0.646081578788237] [1.55627813697339, 0.553876172008416] [1.59931088626975, 0.456908486860592] [1.63259457888104, 0.357501013484366] [1.65685997828737, 0.257932002325746] [1.67308302429485, 0.160292197915408] [1.68238231362625, 0.0663734762396137] [1.68592125935563, -0.0224008565043855] [1.68482630644902, -0.105002516124457] [1.68012745706242, -0.180786774834114] [1.67272231121846, -0.249452742744653] [1.66336100762437, -0.310986527062072] [1.65264731227197, -0.365597542362485] [1.64105051330292, -0.413655999832889] [1.62892327762117, -0.455636888781669] [1.61652168190301, -0.492073246371459] [1.60402482357417, -0.523519544550458] [1.59155248202296, -0.550524701108833] [1.57918011929407, -0.573613477705975] [1.56695106649907, -0.593274723131762] [1.55488607519783, -0.609954906854696] [1.54299057927490, -0.624055540821019] [1.53126006813532, -0.635933315089653] [1.51968396144289, -0.645902016203175] [1.50824833135949, -0.654235522830852] [1.49693776118464, -0.661171365685823] [1.48573657164049, -0.666914493718286] [1.47462959405115, -0.671641008047884] [1.46360262583334, -0.675501713857138] [1.45264266841168, -0.678625404254129] [1.44173802016393, -0.681121834454926] [1.43087827610974, -0.683084374350523] [1.42005427051631, -0.684592346550196] [1.40925798723655, -0.685713068420138] [1.39848245442565, -0.686503622802859] [1.38772163449158, -0.687012384735411] [1.37697031609380, -0.687280331827505] [1.36622401222828, -0.687342164891376] [1.35547886656558, -0.687227263538872] [1.34473156897620, -0.686960499184740] [1.33397928038897, -0.686562925484320] [1.32321956664723, -0.686052363855670] [1.31245034075205, -0.685443899489551] [1.30166981274494, -0.684750301188629] [1.29087644643440, -0.683982376522825] [1.28006892217795, -0.683149272144667] [1.26924610497231, -0.682258727669061] [1.25840701716289, -0.681317290271664] [1.24755081515046, -0.680330496081525] [1.23667676954131, -0.679303023518040] [1.22578424825345, -0.678238822931444] [1.21487270215345, -0.677141226232379] [1.20394165285500, -0.676013039623944] ... [-0.242728623086857, 0.279575839340331] [-0.224841698622698, 0.302590840811220] [-0.205134095734994, 0.325754259098821] [-0.183540590914002, 0.348976621984423] [-0.160002525680198, 0.372153067482177] [-0.134469977024270, 0.395161870316031] [-0.106904238320885, 0.417863032476618] [-0.0772806110015262, 0.440097012012530] [-0.0455914914017819, 0.461683683781087] [-0.0118497160192438, 0.482421647006140] [0.0239078986060691, 0.502088015983098] [0.0616169181949162, 0.520438850291384] [0.101181690554116, 0.537210396794916] [0.142471960915090, 0.552121324054085] [0.185319924941556, 0.564876126252334] [0.229517983154665, 0.575169853482453] [0.274817493846539, 0.582694283243713] [0.320928839394969, 0.587145579917330] [0.367523114991659, 0.588233392191804] [0.414235710785723, 0.585691213353486] [0.460671981323371, 0.579287680988482] [0.506415076240046, 0.568838331585039] [0.551035845193320, 0.554217168584820] [0.594104537120270, 0.535367271912663] [0.635203806507664, 0.512309598462438] [0.673942342664623, 0.485149121249596] [0.709968282219565, 0.454077548753984] [0.742981480671386, 0.419372062483325] [0.772743730135128, 0.381389800885276] [0.799086129190357, 0.340558174331952] [0.821913031412171, 0.297361476157460] [0.841202297595171, 0.252324606583890] [0.857001913278751, 0.205994998255325] [0.869423360156828, 0.158923983550780] [0.878632400961749, 0.111648854141894] [0.884838117000404, 0.0646767355941309] [0.888281107786466, 0.0184711598608366] [0.889221724966838, -0.0265580920222463] [0.887929086978319, -0.0700616376357065] [0.884671436749944, -0.111753702079984] [0.879708196142373, -0.151412212133505] [0.873283868434856, -0.188876276382474] [0.865623767004738, -0.224041647436492] [0.856931417622216, -0.256854783932934] [0.847387397427031, -0.287306093145686] [0.837149332327679, -0.315422858352193] [0.826352768388623, -0.341262257014314] [0.815112651941032, -0.364904771778621] [0.803525187975441, -0.386448197733930] [0.791669888565335, -0.406002363501787] [0.779611666348698, -0.423684614121393] [0.767402868254788, -0.439616051044376] [0.755085179319361, -0.453918487702293] [0.742691354608144, -0.466712055802455] [0.730246758969588, -0.478113385062212] [0.717770710165260, -0.488234274829692] [0.705277631779699, -0.497180777550567] [0.692778029180460, -0.505052619313248] [0.680279305628557, -0.511942890156880] 
WARNING: Output truncated!full_output.txt [1, 1] [1.10000000000000, 0.970000000000000] [1.19554500000000, 0.927804000000000] [1.28497621312080, 0.873492434085960] [1.36680028644537, 0.807637039708032] [1.43979601782619, 0.731316124196504] [1.50310298771150, 0.646081578788237] [1.55627813697339, 0.553876172008416] [1.59931088626975, 0.456908486860592] [1.63259457888104, 0.357501013484366] [1.65685997828737, 0.257932002325746] [1.67308302429485, 0.160292197915408] [1.68238231362625, 0.0663734762396137] [1.68592125935563, -0.0224008565043855] [1.68482630644902, -0.105002516124457] [1.68012745706242, -0.180786774834114] [1.67272231121846, -0.249452742744653] [1.66336100762437, -0.310986527062072] [1.65264731227197, -0.365597542362485] [1.64105051330292, -0.413655999832889] [1.62892327762117, -0.455636888781669] [1.61652168190301, -0.492073246371459] [1.60402482357417, -0.523519544550458] [1.59155248202296, -0.550524701108833] [1.57918011929407, -0.573613477705975] [1.56695106649907, -0.593274723131762] [1.55488607519783, -0.609954906854696] [1.54299057927490, -0.624055540821019] [1.53126006813532, -0.635933315089653] [1.51968396144289, -0.645902016203175] [1.50824833135949, -0.654235522830852] [1.49693776118464, -0.661171365685823] [1.48573657164049, -0.666914493718286] [1.47462959405115, -0.671641008047884] [1.46360262583334, -0.675501713857138] [1.45264266841168, -0.678625404254129] [1.44173802016393, -0.681121834454926] [1.43087827610974, -0.683084374350523] [1.42005427051631, -0.684592346550196] [1.40925798723655, -0.685713068420138] [1.39848245442565, -0.686503622802859] [1.38772163449158, -0.687012384735411] [1.37697031609380, -0.687280331827505] [1.36622401222828, -0.687342164891376] [1.35547886656558, -0.687227263538872] [1.34473156897620, -0.686960499184740] [1.33397928038897, -0.686562925484320] [1.32321956664723, -0.686052363855670] [1.31245034075205, -0.685443899489551] [1.30166981274494, -0.684750301188629] [1.29087644643440, -0.683982376522825] [1.28006892217795, -0.683149272144667] [1.26924610497231, -0.682258727669061] [1.25840701716289, -0.681317290271664] [1.24755081515046, -0.680330496081525] [1.23667676954131, -0.679303023518040] [1.22578424825345, -0.678238822931444] [1.21487270215345, -0.677141226232379] [1.20394165285500, -0.676013039623944] ... [-0.242728623086857, 0.279575839340331] [-0.224841698622698, 0.302590840811220] [-0.205134095734994, 0.325754259098821] [-0.183540590914002, 0.348976621984423] [-0.160002525680198, 0.372153067482177] [-0.134469977024270, 0.395161870316031] [-0.106904238320885, 0.417863032476618] [-0.0772806110015262, 0.440097012012530] [-0.0455914914017819, 0.461683683781087] [-0.0118497160192438, 0.482421647006140] [0.0239078986060691, 0.502088015983098] [0.0616169181949162, 0.520438850291384] [0.101181690554116, 0.537210396794916] [0.142471960915090, 0.552121324054085] [0.185319924941556, 0.564876126252334] [0.229517983154665, 0.575169853482453] [0.274817493846539, 0.582694283243713] [0.320928839394969, 0.587145579917330] [0.367523114991659, 0.588233392191804] [0.414235710785723, 0.585691213353486] [0.460671981323371, 0.579287680988482] [0.506415076240046, 0.568838331585039] [0.551035845193320, 0.554217168584820] [0.594104537120270, 0.535367271912663] [0.635203806507664, 0.512309598462438] [0.673942342664623, 0.485149121249596] [0.709968282219565, 0.454077548753984] [0.742981480671386, 0.419372062483325] [0.772743730135128, 0.381389800885276] [0.799086129190357, 0.340558174331952] [0.821913031412171, 0.297361476157460] [0.841202297595171, 0.252324606583890] [0.857001913278751, 0.205994998255325] [0.869423360156828, 0.158923983550780] [0.878632400961749, 0.111648854141894] [0.884838117000404, 0.0646767355941309] [0.888281107786466, 0.0184711598608366] [0.889221724966838, -0.0265580920222463] [0.887929086978319, -0.0700616376357065] [0.884671436749944, -0.111753702079984] [0.879708196142373, -0.151412212133505] [0.873283868434856, -0.188876276382474] [0.865623767004738, -0.224041647436492] [0.856931417622216, -0.256854783932934] [0.847387397427031, -0.287306093145686] [0.837149332327679, -0.315422858352193] [0.826352768388623, -0.341262257014314] [0.815112651941032, -0.364904771778621] [0.803525187975441, -0.386448197733930] [0.791669888565335, -0.406002363501787] [0.779611666348698, -0.423684614121393] [0.767402868254788, -0.439616051044376] [0.755085179319361, -0.453918487702293] [0.742691354608144, -0.466712055802455] [0.730246758969588, -0.478113385062212] [0.717770710165260, -0.488234274829692] [0.705277631779699, -0.497180777550567] [0.692778029180460, -0.505052619313248] [0.680279305628557, -0.511942890156880]
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