CourseworkGM.ipynb

Kernel: Python 3 (Anaconda 2019)

#Import required modules
import numpy as np
import matplotlib.pyplot as plt
import sympy as sp
from sympy import *
from scipy.integrate import odeint
from scipy import optimize

#Define five second order non-linear differential equations of Lane-Emden Equation
#Represents n=0 to n=5 indexes

def dydt_dzdt(yz,t):
    n = 0
    y = yz[0]
    z = yz[1]
    dydt = z
    dzdt = -(2/t)*z - y**n
    return [dydt, dzdt]
#Define initial conditions
y0 = 1
z0 = 0
#Define t (xi)
t0 = 0.0001
tend = 16
dz = 0.0001
t = np.arange(t0, tend, dz)
#Call odeint
yz = odeint(dydt_dzdt, [y0, z0], t)

x=0.00001
y=1.
u=0.0
h=0.001

while (x<2.5):
    yn=y
    y=y+(h*u)
    u=u+(h*((-(2./x)*u)-(yn)**0))
    x=x+h
    print(x,y)

0.00101 1.0
0.00201 0.999999
0.00301 0.9999989801980198
0.00401 0.9999979800995025
0.00501 0.9999966445182724
0.00601 0.9999949750623441
0.00701 0.9999929720558882
0.00801 0.9999906356073212
0.00901 0.9999879657631955
0.010010000000000002 0.9999849625468166
0.011010000000000002 0.9999816259711433
0.012010000000000003 0.9999779560439561
0.013010000000000004 0.999973952770209
0.014010000000000005 0.9999696161532057
0.015010000000000006 0.9999649461952345
0.016010000000000007 0.9999599428979301
0.017010000000000008 0.9999546062624918
0.01801000000000001 0.999948936289819
0.01901000000000001 0.9999429329805998
0.02001000000000001 0.9999365963353694
0.02101000000000001 0.9999299263545504
0.022010000000000012 0.9999229230384808
0.023010000000000013 0.9999155863874346
0.024010000000000014 0.9999079164016357
0.025010000000000015 0.999899913081269
0.026010000000000016 0.999891576426489
0.027010000000000017 0.9998829064374251
0.028010000000000018 0.9998739031141869
0.02901000000000002 0.9998645664568678
0.03001000000000002 0.999854896465548
0.03101000000000002 0.9998448931402965
0.03201000000000002 0.9998345564811729
0.03301000000000002 0.9998238864882296
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0.03501000000000002 0.9998015465010603
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0.03701000000000002 0.9997778731790917
0.03801000000000002 0.999765536517634
0.039010000000000024 0.9997528665225615
0.040010000000000025 0.9997398631938963
0.041010000000000026 0.9997265265316585
0.042010000000000026 0.999712856535866
0.04301000000000003 0.999698853206535
0.04401000000000003 0.9996845165436801
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0.05701000000000004 0.9994678065806216
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2.333009999999854 0.09323288669719472
2.334009999999854 0.09245521669719595
2.335009999999854 0.09167721336386385
2.336009999999854 0.09089887669719841
2.3370099999998537 0.09012020669719964
2.3380099999998536 0.08934120336386753
2.3390099999998535 0.08856186669720209
2.3400099999998534 0.08778219669720332
2.3410099999998533 0.0870021933638712
2.342009999999853 0.08622185669720576
2.343009999999853 0.08544118669720699
2.344009999999853 0.08466018336387487
2.345009999999853 0.08387884669720942
2.3460099999998527 0.08309717669721065
2.3470099999998526 0.08231517336387853
2.3480099999998525 0.08153283669721308
2.3490099999998524 0.0807501666972143
2.3500099999998523 0.07996716336388218
2.351009999999852 0.07918382669721673
2.352009999999852 0.07840015669721793
2.353009999999852 0.07761615336388582
2.354009999999852 0.07683181669722036
2.3550099999998517 0.07604714669722157
2.3560099999998516 0.07526214336388945
2.3570099999998515 0.074476806697224
2.3580099999998514 0.0736911366972252
2.3590099999998513 0.07290513336389307
2.360009999999851 0.07211879669722761
2.361009999999851 0.07133212669722881
2.362009999999851 0.07054512336389668
2.363009999999851 0.06975778669723122
2.3640099999998507 0.06897011669723242
2.3650099999998506 0.06818211336390029
2.3660099999998505 0.06739377669723481
2.3670099999998504 0.06660510669723602
2.3680099999998503 0.06581610336390388
2.36900999999985 0.0650267666972384
2.37000999999985 0.0642370966972396
2.37100999999985 0.06344709336390747
2.37200999999985 0.062656756697242
2.3730099999998497 0.06186608669724319
2.3740099999998496 0.06107508336391104
2.3750099999998495 0.060283746697245566
2.3760099999998494 0.05949207669724676
2.3770099999998493 0.05870007336391461
2.378009999999849 0.057907736697249135
2.379009999999849 0.05711506669725032
2.380009999999849 0.05632206336391817
2.381009999999849 0.055528726697252694
2.3820099999998487 0.05473505669725388
2.3830099999998486 0.05394105336392173
2.3840099999998485 0.05314671669725624
2.3850099999998484 0.052352046697257425
2.3860099999998483 0.05155704336392527
2.387009999999848 0.05076170669725978
2.388009999999848 0.04996603669726096
2.389009999999848 0.049170033363928806
2.390009999999848 0.04837369669726332
2.3910099999998478 0.04757702669726449
2.3920099999998476 0.04678002336393233
2.3930099999998475 0.045982686697266836
2.3940099999998474 0.04518501669726801
2.3950099999998473 0.04438701336393585
2.396009999999847 0.04358867669727035
2.397009999999847 0.042790006697271524
2.398009999999847 0.04199100336393936
2.399009999999847 0.04119166669727386
2.4000099999998468 0.04039199669727502
2.4010099999998467 0.039591993363942854
2.4020099999998465 0.038791656697277355
2.4030099999998464 0.03799098669727852
2.4040099999998463 0.03718998336394635
2.405009999999846 0.03638864669728084
2.406009999999846 0.035586976697282
2.407009999999846 0.03478497336394983
2.408009999999846 0.033982636697284324
2.4090099999998458 0.033179966697285485
2.4100099999998457 0.03237696336395331
2.4110099999998456 0.0315736266972878
2.4120099999998454 0.030769956697288953
2.4130099999998453 0.029965953363956774
2.4140099999998452 0.02916161669729126
2.415009999999845 0.028356946697292412
2.416009999999845 0.02755194336396023
2.417009999999845 0.026746606697294714
2.418009999999845 0.025940936697295865
2.4190099999998447 0.02513493336396368
2.4200099999998446 0.02432859669729816
2.4210099999998445 0.023521926697299307
2.4220099999998443 0.02271492336396712
2.4230099999998442 0.021907586697301598
2.424009999999844 0.021099916697302743
2.425009999999844 0.020291913363970553
2.426009999999844 0.01948357669730503
2.427009999999844 0.01867490669730617
2.4280099999998437 0.017865903363973975
2.4290099999998436 0.01705656669730845
2.4300099999998435 0.01624689669730959
2.4310099999998434 0.015436893363977394
2.4320099999998432 0.014626556697311863
2.433009999999843 0.013815886697313
2.434009999999843 0.013004883363980802
2.435009999999843 0.012193546697315271
2.436009999999843 0.011381876697316405
2.4370099999998427 0.010569873363984204
2.4380099999998426 0.00975753669731867
2.4390099999998425 0.008944866697319802
2.4400099999998424 0.008131863363987598
2.4410099999998423 0.007318526697322061
2.442009999999842 0.0065048566973231895
2.443009999999842 0.005690853363990984
2.444009999999842 0.004876516697325445
2.445009999999842 0.004061846697326571
2.4460099999998417 0.0032468433639943626
2.4470099999998416 0.0024315066973288205
2.4480099999998415 0.001615836697329944
2.4490099999998414 0.0007998333639977335
2.4500099999998413 -1.650330266781124e-05
2.451009999999841 -0.0008331733026666902
2.452009999999841 -0.0016501766359989034
2.453009999999841 -0.002467513302664451
2.454009999999841 -0.0032851833026633324
2.4550099999998407 -0.004103186635995548
2.4560099999998406 -0.004921523302661098
2.4570099999998405 -0.005740193302659982
2.4580099999998404 -0.0065591966359922005
2.4590099999998403 -0.007378533302657753
2.46000999999984 -0.008198203302656639
2.46100999999984 -0.00901820663598886
2.46200999999984 -0.009838543302654415
2.46300999999984 -0.010659213302653304
2.4640099999998397 -0.011480216635985528
2.4650099999998396 -0.012301553302651085
2.4660099999998395 -0.013123223302649976
2.4670099999998394 -0.013945226635982202
2.4680099999998393 -0.014767563302647762
2.469009999999839 -0.015590233302646656
2.470009999999839 -0.016413236635978885
2.471009999999839 -0.01723657330264445
2.472009999999839 -0.018060243302643345
2.4730099999998387 -0.018884246635975575
2.4740099999998386 -0.01970858330264114
2.4750099999998385 -0.02053325330264004
2.4760099999998384 -0.021358256635972274
2.4770099999998383 -0.022183593302637843
2.478009999999838 -0.023009263302636744
2.479009999999838 -0.02383526663596898
2.480009999999838 -0.02466160330263455
2.481009999999838 -0.025488273302633455
2.4820099999998377 -0.026315276635965695
2.4830099999998376 -0.027142613302631267
2.4840099999998375 -0.027970283302630173
2.4850099999998374 -0.028798286635962414
2.4860099999998373 -0.02962662330262799
2.487009999999837 -0.0304552933026269
2.488009999999837 -0.03128429663595914
2.489009999999837 -0.03211363330262472
2.490009999999837 -0.032943303302623635
2.4910099999998367 -0.03377330663595588
2.4920099999998366 -0.03460364330262146
2.4930099999998365 -0.035434313302620375
2.4940099999998364 -0.03626531663595262
2.4950099999998363 -0.03709665330261821
2.496009999999836 -0.037928323302617126
2.497009999999836 -0.038760326635949374
2.498009999999836 -0.03959266330261496
2.499009999999836 -0.04042533330261388
2.5000099999998358 -0.041258336635946136

def dydt_dadt(ya,t):
    n = 1
    y = ya[0]
    a = ya[1]
    dydt = a
    dadt = -(2/t)*a - y**n
    return [dydt, dadt]
#Define initial conditions
y0 = 1
a0 = 0
#Define t (xi)
t0 = 0.0001
tend = 16
da = 0.0001
t = np.arange(t0, tend, da)
#Call odeint
ya = odeint(dydt_dadt, [y0, a0], t)

x=0.00001
y=1.
u=0.0
h=0.001

while (x<3.5):
    yn=y
    y=y+(h*u)
    u=u+(h*((-(2./x)*u)-(yn)**1))
    x=x+h
    print(x,y)

0.00101 1.0
0.00201 0.999999
0.00301 0.9999989801980198
0.00401 0.9999979801005024
0.00501 0.9999966445206278
0.00601 0.9999949750673988
0.00701 0.9999929720659201
0.00801 0.9999906356256988
0.00901 0.9999879657945657
0.010010000000000002 0.9999849625972996
0.011010000000000002 0.9999816260485308
0.012010000000000003 0.9999779561579101
0.013010000000000004 0.9999739529324608
0.014010000000000005 0.9999696163777564
0.015010000000000006 0.9999649464985539
0.016010000000000007 0.9999599432991574
0.017010000000000008 0.9999546067836347
0.01801000000000001 0.9999489369559541
0.01901000000000001 0.9999429338200725
0.02001000000000001 0.9999365973799936
0.02101000000000001 0.9999299276398088
0.022010000000000012 0.9999229246037248
0.023010000000000013 0.9999155882760837
0.024010000000000014 0.9999079186613783
0.025010000000000015 0.999899915764262
0.026010000000000016 0.9998915795895573
0.027010000000000017 0.9998829101422624
0.028010000000000018 0.9998739074275551
0.02901000000000002 0.9998645714507971
0.03001000000000002 0.9998549022175368
0.03101000000000002 0.9998448997335113
0.03201000000000002 0.9998345640046487
0.03301000000000002 0.9998238950370689
0.03401000000000002 0.9998128928370857
0.03501000000000002 0.9998015574112071
0.03601000000000002 0.9997898887661363
0.03701000000000002 0.9997778869087722
0.03801000000000002 0.9997655518462106
0.039010000000000024 0.9997528835857438
0.040010000000000025 0.9997398821348614
0.041010000000000026 0.999726547501251
0.042010000000000026 0.9997128796927977
0.04301000000000003 0.9996988787175849
0.04401000000000003 0.9996845445838943
0.04501000000000003 0.9996698773002063
0.04601000000000003 0.9996548768751997
0.04701000000000003 0.9996395433177522
0.04801000000000003 0.9996238766369404
0.04901000000000003 0.99960787684204
0.050010000000000034 0.9995915439425256
0.051010000000000034 0.9995748779480711
0.052010000000000035 0.9995578788685495
0.053010000000000036 0.9995405467140333
0.05401000000000004 0.9995228814947938
0.05501000000000004 0.9995048832213024
0.05601000000000004 0.9994865519042292
0.05701000000000004 0.9994678875544442
0.05801000000000004 0.9994488901830165
0.05901000000000004 0.9994295598012151
0.06001000000000004 0.9994098964205079
0.06101000000000004 0.9993899000525629
0.062010000000000044 0.9993695707092471
0.06301000000000004 0.9993489084026276
0.06401000000000004 0.9993279131449704
0.06501000000000004 0.9993065849487415
0.06601000000000004 0.9992849238266063
0.06701000000000004 0.9992629297914297
0.06801000000000004 0.9992406028562764
0.06901000000000004 0.9992179430344102
0.07001000000000004 0.9991949503392952
0.07101000000000005 0.9991716247845942
0.07201000000000005 0.9991479663841701
0.07301000000000005 0.9991239751520854
0.07401000000000005 0.9990996511026018
0.07501000000000005 0.9990749942501809
0.07601000000000005 0.9990500046094836
0.07701000000000005 0.9990246821953703
0.07801000000000005 0.9989990270229013
0.07901000000000005 0.9989730391073361
0.08001000000000005 0.9989467184641339
0.08101000000000005 0.9989200651089531
0.08201000000000006 0.9988930790576522
0.08301000000000006 0.9988657603262886
0.08401000000000006 0.9988381089311196
0.08501000000000006 0.9988101248886017
0.08601000000000006 0.9987818082153911
0.08701000000000006 0.9987531589283434
0.08801000000000006 0.9987241770445136
0.08901000000000006 0.998694862581156
0.09001000000000006 0.9986652155557249
0.09101000000000006 0.9986352359858733
0.09201000000000006 0.9986049238894542
0.09301000000000006 0.9985742792845197
0.09401000000000007 0.9985433021893213
0.09501000000000007 0.9985119926223099
0.09601000000000007 0.998480350602136
0.09701000000000007 0.9984483761476491
0.09801000000000007 0.9984160692778984
0.09901000000000007 0.998383430012132
0.10001000000000007 0.9983504583697979
0.10101000000000007 0.9983171543705428
0.10201000000000007 0.9982835180342131
0.10301000000000007 0.9982495493808544
0.10401000000000007 0.9982152484307115
0.10501000000000008 0.9981806152042284
0.10601000000000008 0.9981456497220486
0.10701000000000008 0.9981103520050145
0.10801000000000008 0.998074722074168
0.10901000000000008 0.99803875995075
0.11001000000000008 0.9980024656562007
0.11101000000000008 0.9979658392121593
0.11201000000000008 0.9979288806404644
0.11301000000000008 0.9978915899631535
0.11401000000000008 0.9978539672024636
0.11501000000000008 0.9978160123808303
0.11601000000000009 0.9977777255208887
0.11701000000000009 0.9977391066454728
0.11801000000000009 0.9977001557776157
0.11901000000000009 0.9976608729405495
0.12001000000000009 0.9976212581577054
0.12101000000000009 0.9975813114527136
0.12201000000000009 0.9975410328494032
0.12301000000000009 0.9975004223718025
0.12401000000000009 0.9974594800441385
0.1250100000000001 0.9974182058908374
0.1260100000000001 0.997376599936524
0.1270100000000001 0.9973346622060225
0.1280100000000001 0.9972923927243555
0.1290100000000001 0.9972497915167449
0.1300100000000001 0.997206858608611
0.1310100000000001 0.9971635940255734
0.1320100000000001 0.9971199977934502
0.1330100000000001 0.9970760699382585
0.1340100000000001 0.997031810486214
0.1350100000000001 0.9969872194637315
0.1360100000000001 0.9969422968974242
0.1370100000000001 0.9968970428141042
0.1380100000000001 0.9968514572407823
0.1390100000000001 0.9968055402046679
0.1400100000000001 0.9967592917331692
0.1410100000000001 0.9967127118538931
0.1420100000000001 0.9966658005946449
0.1430100000000001 0.9966185579834288
0.1440100000000001 0.9965709840484474
0.1450100000000001 0.9965230788181019
0.1460100000000001 0.9964748423209921
0.1470100000000001 0.9964262745859165
0.1480100000000001 0.9963773756418717
0.14901000000000011 0.9963281455180532
0.15001000000000012 0.9962785842438547
0.15101000000000012 0.9962286918488686
0.15201000000000012 0.9961784683628856
0.15301000000000012 0.9961279138158948
0.15401000000000012 0.9960770282380838
0.15501000000000012 0.9960258116598382
0.15601000000000012 0.9959742641117424
0.15701000000000012 0.995922385624579
0.15801000000000012 0.9958701762293289
0.15901000000000012 0.9958176359571712
0.16001000000000012 0.9957647648394832
0.16101000000000013 0.9957115629078406
0.16201000000000013 0.9956580301940173
0.16301000000000013 0.9956041667299854
0.16401000000000013 0.995549972547915
0.16501000000000013 0.9954954476801745
0.16601000000000013 0.9954405921593305
0.16701000000000013 0.9953854060181474
0.16801000000000013 0.9953298892895881
0.16901000000000013 0.9952740420068132
0.17001000000000013 0.9952178642031816
0.17101000000000013 0.99516135591225
0.17201000000000014 0.995104517167773
0.17301000000000014 0.9950473480037035
0.17401000000000014 0.9949898484541921
0.17501000000000014 0.9949320185535875
0.17601000000000014 0.9948738583364359
0.17701000000000014 0.9948153678374818
0.17801000000000014 0.9947565470916673
0.17901000000000014 0.9946973961341323
0.18001000000000014 0.9946379150002148
0.18101000000000014 0.9945781037254501
0.18201000000000014 0.9945179623455714
0.18301000000000014 0.9944574908965099
0.18401000000000015 0.9943966894143942
0.18501000000000015 0.9943355579355505
0.18601000000000015 0.9942740964965029
0.18701000000000015 0.9942123051339729
0.18801000000000015 0.9941501838848795
0.18901000000000015 0.9940877327863397
0.19001000000000015 0.9940249518756675
0.19101000000000015 0.9939618411903749
0.19201000000000015 0.9938984007681709
0.19301000000000015 0.9938346306469623
0.19401000000000015 0.9937705308648531
0.19501000000000016 0.9937061014601449
0.19601000000000016 0.9936413424713365
0.19701000000000016 0.9935762539371241
0.19801000000000016 0.9935108358964013
0.19901000000000016 0.9934450883882588
0.20001000000000016 0.9933790114519848
0.20101000000000016 0.9933126051270645
0.20201000000000016 0.9932458694531805
0.20301000000000016 0.9931788044702123
0.20401000000000016 0.993111410218237
0.20501000000000016 0.9930436867375284
0.20601000000000017 0.9929756340685575
0.20701000000000017 0.9929072522519925
0.20801000000000017 0.9928385413286984
0.20901000000000017 0.9927695013397375
0.21001000000000017 0.9927001323263688
0.21101000000000017 0.9926304343300484
0.21201000000000017 0.9925604073924292
0.21301000000000017 0.9924900515553612
0.21401000000000017 0.9924193668608909
0.21501000000000017 0.992348353351262
0.21601000000000017 0.9922770110689149
0.21701000000000017 0.9922053400564865
0.21801000000000018 0.9921333403568108
0.21901000000000018 0.9920610120129182
0.22001000000000018 0.9919883550680361
0.22101000000000018 0.9919153695655881
0.22201000000000018 0.9918420555491949
0.22301000000000018 0.9917684130626735
0.22401000000000018 0.9916944421500374
0.22501000000000018 0.9916201428554968
0.22601000000000018 0.9915455152234584
0.22701000000000018 0.991470559298525
0.22801000000000018 0.9913952751254963
0.22901000000000019 0.9913196627493681
0.2300100000000002 0.9912437222153327
0.2310100000000002 0.9911674535687788
0.2320100000000002 0.991090856855291
0.2330100000000002 0.9910139321206505
0.2340100000000002 0.9909366794108347
0.2350100000000002 0.9908590987720173
0.2360100000000002 0.9907811902505679
0.2370100000000002 0.9907029538930523
0.2380100000000002 0.9906243897462326
0.2390100000000002 0.9905454978570667
0.2400100000000002 0.9904662782727086
0.2410100000000002 0.9903867310405086
0.2420100000000002 0.9903068562080125
0.2430100000000002 0.9902266538229624
0.2440100000000002 0.990146123933296
0.2450100000000002 0.9900652665871471
0.2460100000000002 0.9899840818328453
0.2470100000000002 0.9899025697189158
0.2480100000000002 0.9898207302940798
0.2490100000000002 0.9897385636072541
0.2500100000000002 0.9896560697075513
0.2510100000000002 0.9895732486442794
0.2520100000000002 0.9894901004669424
0.2530100000000002 0.9894066252252395
0.2540100000000002 0.9893228229690657
0.2550100000000002 0.9892386937485115
0.2560100000000002 0.9891542376138627
0.2570100000000002 0.9890694546156007
0.2580100000000002 0.9889843448044023
0.2590100000000002 0.9888989082311397
0.2600100000000002 0.9888131449468803
0.2610100000000002 0.9887270550028869
0.2620100000000002 0.9886406384506176
0.2630100000000002 0.9885538953417256
0.2640100000000002 0.9884668257280593
0.2650100000000002 0.9883794296616625
0.2660100000000002 0.9882917071947737
0.2670100000000002 0.9882036583798267
0.2680100000000002 0.9881152832694505
0.2690100000000002 0.988026581916469
0.2700100000000002 0.9879375543739009
0.2710100000000002 0.9878482006949598
0.2720100000000002 0.9877585209330545
0.2730100000000002 0.9876685151417884
0.2740100000000002 0.98757818337496
0.2750100000000002 0.9874875256865621
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3.4350099999997328 -0.08439477034398263
3.4360099999997327 -0.08464899242132437
3.4370099999997326 -0.08490298208564992
3.4380099999997324 -0.08515673926109647
3.4390099999997323 -0.08541026387197401
3.4400099999997322 -0.0856635558427653
3.441009999999732 -0.08591661509812595
3.442009999999732 -0.0861694415628844
3.443009999999732 -0.086422035162042
3.444009999999732 -0.08667439582077296
3.4450099999997317 -0.08692652346442446
3.4460099999997316 -0.08717841801851663
3.4470099999997315 -0.08743007940874256
3.4480099999997313 -0.08768150756096839
3.4490099999997312 -0.08793270240123328
3.450009999999731 -0.08818366385574947
3.451009999999731 -0.08843439185090225
3.452009999999731 -0.08868488631325008
3.453009999999731 -0.08893514716952453
3.4540099999997307 -0.08918517434663033
3.4550099999997306 -0.08943496777164542
3.4560099999997305 -0.08968452737182095
3.4570099999997304 -0.08993385307458132
3.4580099999997302 -0.0901829448075242
3.45900999999973 -0.09043180249842053
3.46000999999973 -0.0906804260752146
3.46100999999973 -0.09092881546602403
3.46200999999973 -0.0911769705991398
3.4630099999997297 -0.09142489140302629
3.4640099999997296 -0.09167257780632128
3.4650099999997295 -0.091920029737836
3.4660099999997294 -0.09216724712655516
3.4670099999997293 -0.09241422990163693
3.468009999999729 -0.09266097799241302
3.469009999999729 -0.09290749132838863
3.470009999999729 -0.09315376983924256
3.471009999999729 -0.09339981345482719
3.4720099999997287 -0.09364562210516847
3.4730099999997286 -0.093891195720466
3.4740099999997285 -0.09413653423109304
3.4750099999997284 -0.09438163756759652
3.4760099999997283 -0.09462650566069704
3.477009999999728 -0.09487113844128894
3.478009999999728 -0.0951155358404403
3.479009999999728 -0.09535969778939296
3.480009999999728 -0.09560362421956256
3.4810099999997277 -0.09584731506253852
3.4820099999997276 -0.0960907702500841
3.4830099999997275 -0.09633398971413641
3.4840099999997274 -0.09657697338680643
3.4850099999997273 -0.09681972120037904
3.486009999999727 -0.09706223308731304
3.487009999999727 -0.09730450898024116
3.488009999999727 -0.09754654881197009
3.489009999999727 -0.09778835251548047
3.4900099999997267 -0.09802992002392698
3.4910099999997266 -0.0982712512706383
3.4920099999997265 -0.09851234618911714
3.4930099999997264 -0.0987532047130403
3.4940099999997263 -0.09899382677625863
3.495009999999726 -0.09923421231279711
3.496009999999726 -0.0994743612568548
3.497009999999726 -0.09971427354280493
3.498009999999726 -0.0999539491051949
3.4990099999997257 -0.10019338787874625
3.5000099999997256 -0.10043258979835476

def dydt_dbdt(yb,t):
    n = 2
    y = yb[0]
    b = yb[1]
    dydt = b
    dbdt = -(2/t)*b - y**n
    return [dydt, dbdt]
#Define initial conditions
y0 = 1
b0 = 0
#Define t (xi)
t0 = 0.0001
tend = 16
db = 0.0001
t = np.arange(t0, tend, db)
#Call odeint
yb = odeint(dydt_dbdt, [y0, a0], t)

/ext/anaconda-2019.03/lib/python3.7/site-packages/scipy/integrate/odepack.py:247: ODEintWarning: Excess work done on this call (perhaps wrong Dfun type). Run with full_output = 1 to get quantitative information.
  warnings.warn(warning_msg, ODEintWarning)

x=0.00001
y=1.
u=0.0
h=0.001

while (x<4.5):
    yn=y
    y=y+(h*u)
    u=u+(h*((-(2./x)*u)-(yn)**2))
    x=x+h
    print(x,y)

0.00101 1.0
0.00201 0.999999
0.00301 0.9999989801980198
0.00401 0.9999979801015025
0.00501 0.9999966445229832
0.00601 0.9999949750724534
0.00701 0.9999929720759518
0.00801 0.9999906356440763
0.00901 0.9999879658259357
0.010010000000000002 0.9999849626477824
0.011010000000000002 0.9999816261259177
0.012010000000000003 0.9999779562718627
0.013010000000000004 0.9999739530947105
0.014010000000000005 0.9999696166023033
0.015010000000000006 0.9999649468018674
0.016010000000000007 0.9999599437003753
0.017010000000000008 0.9999546073047635
0.01801000000000001 0.9999489376220686
0.01901000000000001 0.9999429346595157
0.02001000000000001 0.9999365984245766
0.02101000000000001 0.9999299289250105
0.022010000000000012 0.9999229261688919
0.023010000000000013 0.9999155901646305
0.024010000000000014 0.9999079209209861
0.025010000000000015 0.9998999184470795
0.026010000000000016 0.9998915827524002
0.027010000000000017 0.9998829138468127
0.028010000000000018 0.9998739117405614
0.02901000000000002 0.9998645764442741
0.03001000000000002 0.9998549079689647
0.03101000000000002 0.999844906326036
0.03201000000000002 0.9998345715272808
0.03301000000000002 0.999823903584884
0.03401000000000002 0.9998129025114235
0.03501000000000002 0.999801568319871
0.03601000000000002 0.999789901023593
0.03701000000000002 0.9997779006363513
0.03801000000000002 0.9997655671723038
0.039010000000000024 0.9997529006460046
0.040010000000000025 0.9997399010724045
0.041010000000000026 0.9997265684668514
0.042010000000000026 0.9997129028450902
0.04301000000000003 0.9996989042232634
0.04401000000000003 0.999684572617911
0.04501000000000003 0.999669908045971
0.04601000000000003 0.9996549105247791
0.04701000000000003 0.9996395800720687
0.04801000000000003 0.9996239167059715
0.04901000000000003 0.9996079204450173
0.050010000000000034 0.9995915913081339
0.051010000000000034 0.9995749293146471
0.052010000000000035 0.9995579344842811
0.053010000000000036 0.9995406068371581
0.05401000000000004 0.9995229463937985
0.05501000000000004 0.9995049531751209
0.05601000000000004 0.9994866272024422
0.05701000000000004 0.9994679684974771
0.05801000000000004 0.9994489770823386
0.05901000000000004 0.9994296529795378
0.06001000000000004 0.9994099962119838
0.06101000000000004 0.9993900068029837
0.062010000000000044 0.9993696847762428
0.06301000000000004 0.9993490301558638
0.06401000000000004 0.9993280429663479
0.06501000000000004 0.9993067232325938
0.06601000000000004 0.999285070979898
0.06701000000000004 0.999263086233955
0.06801000000000004 0.9992407690208566
0.06901000000000004 0.9992181193670924
0.07001000000000004 0.9991951372995497
0.07101000000000005 0.9991718228455132
0.07201000000000005 0.999148176032665
0.07301000000000005 0.9991241968890848
0.07401000000000005 0.9990998854432492
0.07501000000000005 0.9990752417240325
0.07601000000000005 0.9990502657607059
0.07701000000000005 0.9990249575829377
0.07801000000000005 0.9989993172207936
0.07901000000000005 0.9989733447047358
0.08001000000000005 0.9989470400656235
0.08101000000000005 0.998920403334713
0.08201000000000006 0.998893434543657
0.08301000000000006 0.998866133724505
0.08401000000000006 0.998838500909703
0.08501000000000006 0.9988105361320937
0.08601000000000006 0.998782239424916
0.08701000000000006 0.9987536108218051
0.08801000000000006 0.9987246503567928
0.08901000000000006 0.9986953580643066
0.09001000000000006 0.9986657339791704
0.09101000000000006 0.9986357781366038
0.09201000000000006 0.9986054905722225
0.09301000000000006 0.9985748713220379
0.09401000000000007 0.9985439204224573
0.09501000000000007 0.9985126379102834
0.09601000000000007 0.9984810238227145
0.09701000000000007 0.9984490781973443
0.09801000000000007 0.9984168010721618
0.09901000000000007 0.9983841924855512
0.10001000000000007 0.9983512524762921
0.10101000000000007 0.9983179810835587
0.10201000000000007 0.9982843783469206
0.10301000000000007 0.9982504443063419
0.10401000000000007 0.9982161790021816
0.10501000000000008 0.9981815824751932
0.10601000000000008 0.9981466547665248
0.10701000000000008 0.9981113959177187
0.10801000000000008 0.9980758059707119
0.10901000000000008 0.9980398849678352
0.11001000000000008 0.9980036329518139
0.11101000000000008 0.9979670499657669
0.11201000000000008 0.9979301360532071
0.11301000000000008 0.9978928912580413
0.11401000000000008 0.9978553156245695
0.11501000000000008 0.9978174091974857
0.11601000000000009 0.9977791720218773
0.11701000000000009 0.9977406041432245
0.11801000000000009 0.9977017056074012
0.11901000000000009 0.9976624764606741
0.12001000000000009 0.9976229167497028
0.12101000000000009 0.9975830265215399
0.12201000000000009 0.9975428058236304
0.12301000000000009 0.9975022547038124
0.12401000000000009 0.9974613732103158
0.1250100000000001 0.9974201613917633
0.1260100000000001 0.9973786192971694
0.1270100000000001 0.9973367469759411
0.1280100000000001 0.997294544477877
0.1290100000000001 0.9972520118531677
0.1300100000000001 0.9972091491523954
0.1310100000000001 0.9971659564265339
0.1320100000000001 0.9971224337269483
0.1330100000000001 0.997078581105395
0.1340100000000001 0.9970343986140218
0.1350100000000001 0.9969898863053671
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0.1400100000000001 0.9967623793690783
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0.1460100000000001 0.9964784983866304
0.1470100000000001 0.9964300325107516
0.1480100000000001 0.9963812375361124
0.14901000000000011 0.9963321135207108
0.15001000000000012 0.9962826605229338
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0.15201000000000012 0.9961827678157446
0.15301000000000012 0.996132328225049
0.15401000000000012 0.9960815598894114
0.15501000000000012 0.9960304628691605
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0.15801000000000012 0.9958752003098365
0.15901000000000012 0.9958227891621788
0.16001000000000012 0.9957700496373687
0.16101000000000013 0.9957169817980602
0.16201000000000013 0.9956635857072947
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0.16401000000000013 0.9955558090254915
0.16501000000000013 0.9955014285624694
0.16601000000000013 0.9954467201040217
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0.16901000000000013 0.9952806274077907
0.17001000000000013 0.9952246076212358
0.17101000000000013 0.9951682601679817
0.17201000000000014 0.9951115851149301
0.17301000000000014 0.9950545825293683
0.17401000000000014 0.9949972524789682
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0.17601000000000014 0.9948816102562646
0.17701000000000014 0.9948232982212284
0.17801000000000014 0.9947646589958877
0.17901000000000014 0.9947056926498369
0.18001000000000014 0.9946463992530539
0.18101000000000014 0.9945867788759004
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0.18401000000000015 0.9944059565715868
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4.413009999999809 -0.007725364689915824
4.414009999999809 -0.007849143425427366
4.41500999999981 -0.007972866123427772
4.41600999999981 -0.008096532823944477
4.41700999999981 -0.008220143566999277
4.418009999999811 -0.00834369839260831
4.419009999999811 -0.008467197340782044
4.420009999999811 -0.008590640451525258
4.421009999999812 -0.008714027764837035
4.422009999999812 -0.00883735932071074
4.423009999999812 -0.008960635159134013
4.424009999999813 -0.009083855320088749
4.425009999999813 -0.009207019843551089
4.426009999999813 -0.009330128769491401
4.427009999999814 -0.00945318213787427
4.428009999999814 -0.009576179988658484
4.429009999999814 -0.009699122361797015
4.430009999999815 -0.009822009297237012
4.431009999999815 -0.009944840834919783
4.432009999999815 -0.01006761701478078
4.433009999999816 -0.01019033787674959
4.434009999999816 -0.01031300346074992
4.435009999999816 -0.010435613806699578
4.436009999999817 -0.010558168954510467
4.437009999999817 -0.010680668944088567
4.438009999999817 -0.01080311381533392
4.439009999999818 -0.01092550360814062
4.440009999999818 -0.011047838362396803
4.441009999999818 -0.011170118117984622
4.442009999999819 -0.011292342914780243
4.443009999999819 -0.011414512792653831
4.444009999999819 -0.011536627791469535
4.44500999999982 -0.01165868795108547
4.44600999999982 -0.011780693311353712
4.44700999999982 -0.011902643912120281
4.448009999999821 -0.012024539793225129
4.449009999999821 -0.012146380994502121
4.450009999999821 -0.012268167555779034
4.451009999999822 -0.012389899516877531
4.452009999999822 -0.012511576917613157
4.453009999999822 -0.012633199797795322
4.454009999999823 -0.012754768197227288
4.455009999999823 -0.012876282155706159
4.456009999999823 -0.012997741713022865
4.457009999999824 -0.013119146908962149
4.458009999999824 -0.013240497783302559
4.459009999999824 -0.013361794375816428
4.460009999999825 -0.01348303672626987
4.461009999999825 -0.013604224874422756
4.462009999999825 -0.013725358860028717
4.463009999999826 -0.013846438722835114
4.464009999999826 -0.013967464502583038
4.465009999999826 -0.014088436239007294
4.466009999999827 -0.014209353971836385
4.467009999999827 -0.014330217740792505
4.468009999999827 -0.014451027585591523
4.469009999999828 -0.014571783545942973
4.470009999999828 -0.01469248566155004
4.471009999999828 -0.014813133972109548
4.472009999999829 -0.014933728517311947
4.473009999999829 -0.015054269336841303
4.474009999999829 -0.015174756470375287
4.47500999999983 -0.015295189957585156
4.47600999999983 -0.015415569838135749
4.47700999999983 -0.01553589615168547
4.478009999999831 -0.015656168937886277
4.479009999999831 -0.015776388236383675
4.480009999999831 -0.015896554086816693
4.481009999999832 -0.016016666528817884
4.482009999999832 -0.0161367256020133
4.483009999999832 -0.016256731346022502
4.484009999999833 -0.016376683800458522
4.485009999999833 -0.016496583004927866
4.486009999999833 -0.016616428999030507
4.487009999999834 -0.016736221822359856
4.488009999999834 -0.01685596151450277
4.489009999999834 -0.016975648115039522
4.490009999999835 -0.017095281663543807
4.491009999999835 -0.017214862199582716
4.492009999999835 -0.017334389762716732
4.493009999999836 -0.01745386439249972
4.494009999999836 -0.01757328612847891
4.495009999999836 -0.017692655010194885
4.496009999999837 -0.01781197107718158
4.497009999999837 -0.01793123436896626
4.498009999999837 -0.018050444925069516
4.499009999999838 -0.018169602785005245
4.500009999999838 -0.018288707988280646

def dydt_dcdt(yc,t):
    n = 3
    y = yc[0]
    c = yc[1]
    dydt = c
    dcdt = -(2/t)*c - y**n
    return [dydt, dcdt]
#Define initial conditions
y0 = 1
c0 = 0
#Define t (xi)
t0 = 0.0001
tend = 16
dc = 0.0001
t = np.arange(t0, tend, dc)
#Call odeint
yc = odeint(dydt_dcdt, [y0, a0], t)

x=0.00001
y=1.
u=0.0
h=0.001

while (x<7):
    yn=y
    y=y+(h*u)
    u=u+(h*((-(2./x)*u)-(yn)**3))
    x=x+h
    print(x,y)

WARNING: some intermediate output was truncated.
WARNING: 2 intermediate output messages were discarded.
WARNING: some intermediate output was truncated.
WARNING: some intermediate output was truncated.

 0.6657022588272449
1.6940099999999243 0.6654255160996938
1.6950099999999242 0.665148805283778
1.696009999999924 0.6648721265166039
1.697009999999924 0.6645954799350403
1.6980099999999239 0.6643188656757185
1.6990099999999237 0.6640422838750323
1.7000099999999236 0.6637657346691379
1.7010099999999235 0.6634892181939547
1.7020099999999234 0.6632127345851646
1.7030099999999233 0.662936283978213
1.7040099999999232 0.6626598665083084
1.705009999999923 0.6623834823104227
1.706009999999923 0.6621071315192909
1.7070099999999229 0.6618308142694123
1.7080099999999228 0.6615545306950494
1.7090099999999226 0.661278280930229
1.7100099999999225 0.6610020651087418
1.7110099999999224 0.6607258833641428
1.7120099999999223 0.6604497358297513
1.7130099999999222 0.660173622638651
1.714009999999922 0.6598975439236904
1.715009999999922 0.6596214998174826
1.7160099999999219 0.6593454904524059
1.7170099999999218 0.6590695159606033
1.7180099999999217 0.6587935764739833
1.7190099999999215 0.6585176721242197
1.7200099999999214 0.6582418030427518
1.7210099999999213 0.6579659693607844
1.7220099999999212 0.6576901712092883
1.723009999999921 0.6574144087190001
1.724009999999921 0.6571386820204227
1.7250099999999209 0.656862991243825
1.7260099999999208 0.6565873365192424
1.7270099999999207 0.6563117179764768
1.7280099999999206 0.6560361357450968
1.7290099999999204 0.6557605899544379
1.7300099999999203 0.6554850807336025
1.7310099999999202 0.6552096082114601
1.7320099999999201 0.6549341725166477
1.73300999999992 0.6546587737775694
1.73400999999992 0.6543834121223973
1.7350099999999198 0.6541080876790708
1.7360099999999197 0.6538328005752976
1.7370099999999196 0.6535575509385532
1.7380099999999195 0.6532823388960813
1.7390099999999193 0.6530071645748942
1.7400099999999192 0.6527320281017722
1.7410099999999191 0.6524569296032648
1.742009999999919 0.6521818692056902
1.743009999999919 0.6519068470351352
1.7440099999999188 0.6516318632174563
1.7450099999999187 0.6513569178782787
1.7460099999999186 0.6510820111429975
1.7470099999999185 0.6508071431367771
1.7480099999999184 0.6505323139845519
1.7490099999999182 0.650257523811026
1.7500099999999181 0.6499827727406736
1.751009999999918 0.6497080608977392
1.752009999999918 0.6494333884062378
1.7530099999999178 0.6491587553899545
1.7540099999999177 0.6488841619724456
1.7550099999999176 0.6486096082770381
1.7560099999999175 0.6483350944268298
1.7570099999999174 0.64806062054469
1.7580099999999172 0.6477861867532593
1.7590099999999171 0.6475117931749494
1.760009999999917 0.6472374399319444
1.761009999999917 0.6469631271461994
1.7620099999999168 0.6466888549394422
1.7630099999999167 0.6464146234331722
1.7640099999999166 0.6461404327486616
1.7650099999999165 0.6458662830069547
1.7660099999999164 0.6455921743288685
1.7670099999999163 0.645318106834993
1.7680099999999161 0.645044080645691
1.769009999999916 0.6447700958810985
1.770009999999916 0.6444961526611247
1.7710099999999158 0.6442222511054524
1.7720099999999157 0.643948391333538
1.7730099999999156 0.6436745734646117
1.7740099999999155 0.6434007976176774
1.7750099999999154 0.6431270639115136
1.7760099999999153 0.6428533724646729
1.7770099999999152 0.6425797233954821
1.778009999999915 0.6423061168220429
1.779009999999915 0.6420325528622317
1.7800099999999148 0.6417590316337
1.7810099999999147 0.6414855532538742
1.7820099999999146 0.6412121178399564
1.7830099999999145 0.6409387255089236
1.7840099999999144 0.6406653763775289
1.7850099999999143 0.640392070562301
1.7860099999999142 0.6401188081795446
1.787009999999914 0.6398455893453407
1.788009999999914 0.6395724141755466
1.7890099999999138 0.6392992827857957
1.7900099999999137 0.6390261952914986
1.7910099999999136 0.6387531518078424
1.7920099999999135 0.6384801524497914
1.7930099999999134 0.6382071973320869
1.7940099999999133 0.6379342865692477
1.7950099999999132 0.6376614202755699
1.796009999999913 0.6373885985651278
1.797009999999913 0.6371158215517729
1.7980099999999128 0.6368430893491354
1.7990099999999127 0.6365704020706231
1.8000099999999126 0.6362977598294227
1.8010099999999125 0.6360251627384992
1.8020099999999124 0.6357526109105964
1.8030099999999123 0.6354801044582371
1.8040099999999122 0.635207643493723
1.805009999999912 0.6349352281291354
1.806009999999912 0.6346628584763346
1.8070099999999119 0.6343905346469609
1.8080099999999117 0.6341182567524343
1.8090099999999116 0.6338460249039547
1.8100099999999115 0.6335738392125021
1.8110099999999114 0.6333016997888371
1.8120099999999113 0.6330296067435005
1.8130099999999112 0.632757560186814
1.814009999999911 0.6324855602288799
1.815009999999911 0.632213606979582
1.8160099999999109 0.6319417005485849
1.8170099999999108 0.6316698410453347
1.8180099999999106 0.6313980285790594
1.8190099999999105 0.6311262632587682
1.8200099999999104 0.6308545451932527
1.8210099999999103 0.6305828744910864
1.8220099999999102 0.6303112512606253
1.82300999999991 0.6300396756100075
1.82400999999991 0.6297681476471542
1.8250099999999099 0.6294966674797691
1.8260099999999098 0.6292252352153391
1.8270099999999097 0.6289538509611344
1.8280099999999095 0.6286825148242084
1.8290099999999094 0.6284112269113983
1.8300099999999093 0.6281399873293246
1.8310099999999092 0.6278687961843923
1.832009999999909 0.6275976535827902
1.833009999999909 0.6273265596304913
1.8340099999999089 0.6270555144332535
1.8350099999999088 0.626784518096619
1.8360099999999087 0.626513570725915
1.8370099999999085 0.6262426724262536
1.8380099999999084 0.6259718233025324
1.8390099999999083 0.6257010234594342
1.8400099999999082 0.6254302730014274
1.841009999999908 0.6251595720327662
1.842009999999908 0.6248889206574908
1.8430099999999079 0.6246183189794277
1.8440099999999078 0.6243477671021893
1.8450099999999077 0.6240772651291749
1.8460099999999076 0.6238068131635705
1.8470099999999074 0.6235364113083487
1.8480099999999073 0.6232660596662696
1.8490099999999072 0.6229957583398802
1.8500099999999071 0.6227255074315151
1.851009999999907 0.6224553070432965
1.852009999999907 0.6221851572771346
1.8530099999999068 0.6219150582347274
1.8540099999999067 0.6216450100175611
1.8550099999999066 0.6213750127269105
1.8560099999999065 0.621105066463839
1.8570099999999063 0.6208351713291985
1.8580099999999062 0.62056532742363
1.8590099999999061 0.6202955348475637
1.860009999999906 0.6200257937012191
1.861009999999906 0.6197561040846052
1.8620099999999058 0.6194864660975208
1.8630099999999057 0.6192168798395544
1.8640099999999056 0.618947345410085
1.8650099999999055 0.6186778629082815
1.8660099999999054 0.6184084324331035
1.8670099999999052 0.618139054083301
1.8680099999999051 0.6178697279574153
1.869009999999905 0.6176004541537785
1.870009999999905 0.6173312327705138
1.8710099999999048 0.6170620639055361
1.8720099999999047 0.6167929476565518
1.8730099999999046 0.6165238841210592
1.8740099999999045 0.6162548733963484
1.8750099999999044 0.6159859155795021
1.8760099999999043 0.615717010767395
1.8770099999999041 0.6154481590566947
1.878009999999904 0.6151793605438612
1.879009999999904 0.614910615325148
1.8800099999999038 0.6146419234966013
1.8810099999999037 0.6143732851540609
1.8820099999999036 0.61410470039316
1.8830099999999035 0.6138361693093258
1.8840099999999034 0.6135676919977794
1.8850099999999033 0.6132992685535357
1.8860099999999032 0.6130308990714043
1.887009999999903 0.6127625836459891
1.888009999999903 0.612494322371689
1.8890099999999028 0.6122261153426976
1.8900099999999027 0.6119579626530035
1.8910099999999026 0.611689864396391
1.8920099999999025 0.6114218206664395
1.8930099999999024 0.6111538315565243
1.8940099999999023 0.6108858971598166
1.8950099999999022 0.6106180175692836
1.896009999999902 0.6103501928776889
1.897009999999902 0.6100824231775925
1.8980099999999018 0.6098147085613512
1.8990099999999017 0.6095470491211186
1.9000099999999016 0.6092794449488451
1.9010099999999015 0.6090118961362789
1.9020099999999014 0.6087444027749653
1.9030099999999013 0.6084769649562474
1.9040099999999012 0.608209582771266
1.905009999999901 0.6079422563109602
1.906009999999901 0.6076749856660673
1.9070099999999008 0.6074077709271228
1.9080099999999007 0.6071406121844612
1.9090099999999006 0.6068735095282157
1.9100099999999005 0.6066064630483184
1.9110099999999004 0.606339472834501
1.9120099999999003 0.6060725389762942
1.9130099999999002 0.6058056615630287
1.9140099999999 0.605538840683835
1.9150099999999 0.6052720764276436
1.9160099999998998 0.6050053688831851
1.9170099999998997 0.6047387181389907
1.9180099999998996 0.6044721242833923
1.9190099999998995 0.6042055874045222
1.9200099999998994 0.6039391075903144
1.9210099999998993 0.6036726849285038
1.9220099999998992 0.6034063195066266
1.923009999999899 0.6031400114120209
1.924009999999899 0.6028737607318265
1.9250099999998989 0.6026075675529854
1.9260099999998987 0.6023414319622415
1.9270099999998986 0.6020753540461415
1.9280099999998985 0.6018093338910346
1.9290099999998984 0.6015433715830728
1.9300099999998983 0.6012774672082112
1.9310099999998982 0.601011620852208
1.932009999999898 0.6007458326006252
1.933009999999898 0.6004801025388281
1.9340099999998979 0.600214430751986
1.9350099999998978 0.5999488173250721
1.9360099999998976 0.5996832623428642
1.9370099999998975 0.5994177658899441
1.9380099999998974 0.5991523280506986
1.9390099999998973 0.5988869489093193
1.9400099999998972 0.5986216285498027
1.941009999999897 0.5983563670559507
1.942009999999897 0.5980911645113707
1.9430099999998969 0.5978260209994756
1.9440099999998968 0.5975609366034842
1.9450099999998967 0.5972959114064216
1.9460099999998965 0.5970309454911188
1.9470099999998964 0.5967660389402136
1.9480099999998963 0.5965011918361505
1.9490099999998962 0.5962364042611805
1.950009999999896 0.5959716762973621
1.951009999999896 0.595707008026561
1.9520099999998959 0.5954423995304501
1.9530099999998958 0.5951778508905103
1.9540099999998957 0.5949133621880305
1.9550099999998956 0.5946489335041073
1.9560099999998954 0.5943845649196459
1.9570099999998953 0.5941202565153599
1.9580099999998952 0.5938560083717718
1.9590099999998951 0.5935918205692128
1.960009999999895 0.5933276931878232
1.961009999999895 0.5930636263075528
1.9620099999998948 0.592799620008161
1.9630099999998947 0.5925356743692167
1.9640099999998946 0.5922717894700988
1.9650099999998945 0.5920079653899964
1.9660099999998943 0.591744202207909
1.9670099999998942 0.5914805000026466
1.9680099999998941 0.5912168588528299
1.969009999999894 0.5909532788368906
1.970009999999894 0.5906897600330717
1.9710099999998938 0.5904263025194274
1.9720099999998937 0.5901629063738235
1.9730099999998936 0.5898995716739377
1.9740099999998935 0.5896362984972596
1.9750099999998934 0.589373086921091
1.9760099999998932 0.5891099370225462
1.9770099999998931 0.588846848878552
1.978009999999893 0.588583822565848
1.979009999999893 0.5883208581609869
1.9800099999998928 0.5880579557403346
1.9810099999998927 0.5877951153800703
1.9820099999998926 0.5875323371561871
1.9830099999998925 0.5872696211444918
1.9840099999998924 0.5870069674206051
1.9850099999998922 0.5867443760599622
1.9860099999998921 0.5864818471378127
1.987009999999892 0.5862193807292208
1.988009999999892 0.5859569769090655
1.9890099999998918 0.585694635752041
1.9900099999998917 0.5854323573326567
1.9910099999998916 0.5851701417252378
1.9920099999998915 0.5849079890039248
1.9930099999998914 0.5846458992426742
1.9940099999998913 0.5843838725152587
1.9950099999998911 0.5841219088952673
1.996009999999891 0.5838600084561055
1.997009999999891 0.5835981712709956
1.9980099999998908 0.5833363974129767
1.9990099999998907 0.5830746869549052
2.000009999999891 0.5828130399694547
2.0010099999998907 0.5825514565291164
2.0020099999998906 0.5822899367061994
2.0030099999998905 0.5820284805728307
2.0040099999998904 0.5817670882009555
2.0050099999998903 0.5815057596623371
2.00600999999989 0.5812444950285578
2.00700999999989 0.5809832943710186
2.00800999999989 0.5807221577609393
2.00900999999989 0.5804610852693591
2.0100099999998897 0.5802000769671367
2.0110099999998896 0.5799391329249503
2.0120099999998895 0.5796782532132978
2.0130099999998894 0.5794174379024976
2.0140099999998893 0.5791566870626877
2.015009999999889 0.5788960007638273
2.016009999999889 0.5786353790756956
2.017009999999889 0.5783748220678931
2.018009999999889 0.5781143298098412
2.0190099999998887 0.5778539023707827
2.0200099999998886 0.5775935398197818
2.0210099999998885 0.5773332422257245
2.0220099999998884 0.5770730096573186
2.0230099999998883 0.5768128421830941
2.024009999999888 0.5765527398714034
2.025009999999888 0.5762927027904213
2.026009999999888 0.5760327310081454
2.027009999999888 0.5757728245923964
2.0280099999998877 0.5755129836108179
2.0290099999998876 0.575253208130877
2.0300099999998875 0.5749934982198643
2.0310099999998874 0.5747338539448945
2.0320099999998873 0.5744742753729059
2.033009999999887 0.5742147625706611
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6.995010000000671 -0.004190181346291915
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6.997010000000672 -0.004272612892760039
6.998010000000672 -0.004313810991904578
6.9990100000006725 -0.004354997315055651
7.000010000000673 -0.004396171867259259

WARNING: 2 intermediate output messages were discarded.
WARNING: some intermediate output was truncated.
WARNING: some intermediate output was truncated.
WARNING: some intermediate output was truncated.

def dydt_dedt(ye,t):
    n = 4
    y = ye[0]
    e = ye[1]
    dydt = e
    dedt = -(2/t)*e - y**n
    return [dydt, dedt]
#Define initial conditions
y0 = 1
e0 = 0
#Define t (xi)
t0 = 0.0001
tend = 16
de = 0.0001
t = np.arange(t0, tend, de)
#Call odeint
ye = odeint(dydt_dedt, [y0, e0], t)

x=0.00001
y=1.
u=0.0
h=0.001

while (x<15.5):
    yn=y
    y=y+(h*u)
    u=u+(h*((-(2./x)*u)-(yn)**4))
    x=x+h
    print(x,y)

WARNING: some intermediate output was truncated.
WARNING: 11 intermediate output messages were discarded.

9.94300999999993 0.06069624481841043
9.94400999999993 0.060678092310938396
9.945009999999929 0.06065994344120457
9.946009999999928 0.06064179820812634
9.947009999999928 0.06062365661062153
9.948009999999927 0.06060551864760838
9.949009999999927 0.060587384318005554
9.950009999999926 0.06056925362073213
9.951009999999926 0.0605511265547076
9.952009999999925 0.060533003118851895
9.953009999999924 0.06051488331208535
9.954009999999924 0.06049676713332872
9.955009999999923 0.0604786545815032
9.956009999999923 0.06046054565553037
9.957009999999922 0.06044244035433225
9.958009999999922 0.060424338676831287
9.959009999999921 0.060406240621950325
9.96000999999992 0.060388146188612636
9.96100999999992 0.06037005537574191
9.96200999999992 0.06035196818226226
9.963009999999919 0.06033388460709821
9.964009999999918 0.0603158046491747
9.965009999999918 0.060297728307417094
9.966009999999917 0.06027965558075117
9.967009999999917 0.060261586468103125
9.968009999999916 0.06024352096839957
9.969009999999916 0.060225459080567534
9.970009999999915 0.060207400803534455
9.971009999999914 0.0601893461362282
9.972009999999914 0.06017129507757705
9.973009999999913 0.060153247626509684
9.974009999999913 0.06013520378195522
9.975009999999912 0.06011716354284318
9.976009999999912 0.060099126908103505
9.977009999999911 0.06008109387666655
9.97800999999991 0.06006306444746307
9.97900999999991 0.060045038619424264
9.98000999999991 0.060027016391481725
9.981009999999909 0.06000899776256746
9.982009999999908 0.0599909827316139
9.983009999999908 0.05997297129755388
9.984009999999907 0.05995496345932065
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9.989009999999904 0.05986497816824922
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9.9960099999999 0.05973914948727976
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10.000009999999898 0.05966732621116774
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10.003009999999897 0.059613496324225954
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10.005009999999896 0.05957762760758838
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10.009009999999893 0.05950593303540919
10.010009999999893 0.05948801831559485
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10.012009999999892 0.05945219957645173
10.013009999999891 0.05943429555501287
10.01400999999989 0.05941639509758976
10.01500999999989 0.05939849820312839
10.01600999999989 0.05938060487057514
10.017009999999889 0.05936271509887681
10.018009999999888 0.0593448288869806
10.019009999999888 0.05932694623383412
10.020009999999887 0.05930906713838537
10.021009999999887 0.05929119159958279
10.022009999999886 0.0592733196163752
10.023009999999886 0.05925545118771183
10.024009999999885 0.05923758631254233
10.025009999999885 0.05921972498981673
10.026009999999884 0.05920186721848549
10.027009999999883 0.059184012997499466
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10.029009999999882 0.059148315202368504
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10.031009999999881 0.059112631596038794
10.03200999999988 0.05909479511105585
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10.03400999999988 0.059059132772220205
10.035009999999879 0.05904130691627529
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10.037009999999878 0.059005665826103726
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10.041009999999876 0.058934426103385
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10.044009999999874 0.05888103342889423
10.045009999999873 0.05886324293517939
10.046009999999873 0.05884545597195279
10.047009999999872 0.05882767253817335
10.048009999999872 0.058809892632800405
10.049009999999871 0.05879211625479367
10.05000999999987 0.05877434340311328
10.05100999999987 0.058756574076719754
10.05200999999987 0.05873880827457403
10.053009999999869 0.058721045995637426
10.054009999999868 0.05870328723887168
10.055009999999868 0.058685532003238915
10.056009999999867 0.058667780287701664
10.057009999999867 0.05865003209122285
10.058009999999866 0.05863228741276581
10.059009999999866 0.05861454625129427
10.060009999999865 0.05859680860577234
10.061009999999865 0.05857907447516456
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10.063009999999863 0.058543616754551546
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10.074009999999857 0.058348850164034856
10.075009999999857 0.05833116513372327
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15.453009999996876 -0.003696053789855485
15.454009999996876 -0.003703573151831014
15.455009999996875 -0.003711091540616137
15.456009999996875 -0.003718608956399776
15.457009999996874 -0.003726125399370804
15.458009999996873 -0.0037336408697180444
15.459009999996873 -0.003741155367630272
15.460009999996872 -0.003748668893296213
15.461009999996872 -0.003756181446904545
15.462009999996871 -0.0037636930286438957
15.46300999999687 -0.0037712036387028453
15.46400999999687 -0.0037787132772699244
15.46500999999687 -0.003786221944533615
15.466009999996869 -0.003793729640682351
15.467009999996868 -0.0038012363659045167
15.468009999996868 -0.003808742120388448
15.469009999996867 -0.0038162469043224317
15.470009999996867 -0.003823750717894707
15.471009999996866 -0.0038312535612934637
15.472009999996866 -0.0038387554347068427
15.473009999996865 -0.0038462563383229367
15.474009999996865 -0.0038537562723297896
15.475009999996864 -0.003861255236915397
15.476009999996863 -0.003868753232267706
15.477009999996863 -0.003876250258574614
15.478009999996862 -0.003883746316023971
15.479009999996862 -0.0038912414048035784
15.480009999996861 -0.0038987355251011886
15.48100999999686 -0.0039062286771045056
15.48200999999686 -0.003913720861001185
15.48300999999686 -0.003921212076978834
15.484009999996859 -0.003928702325225012
15.485009999996858 -0.003936191605927228
15.486009999996858 -0.003943679919272944
15.487009999996857 -0.003951167265449573
15.488009999996857 -0.003958653644644481
15.489009999996856 -0.003966139057044985
15.490009999996856 -0.003973623502838352
15.491009999996855 -0.003981106982211801
15.492009999996855 -0.003988589495352506
15.493009999996854 -0.003996071042447589
15.494009999996853 -0.004003551623684125
15.495009999996853 -0.004011031239249141
15.496009999996852 -0.004018509889329615
15.497009999996852 -0.004025987574112476
15.498009999996851 -0.004033464293784607
15.49900999999685 -0.0040409400485328425
15.50000999999685 -0.004048414838543966

WARNING: 11 intermediate output messages were discarded.
WARNING: some intermediate output was truncated.

def dydt_dpdt(yp,t):
    n = 5
    y = yp[0]
    p = yp[1]
    dydt = p
    dpdt = -(2/t)*p - y**n
    return [dydt, dpdt]
#Define initial conditions
y0 = 1
p0 = 0
#Define t (xi)
t0 = 0.0001
tend = 16
dp = 0.0001
t = np.arange(t0, tend, dp)
#Call odeint
yp = odeint(dydt_dpdt, [y0, p0], t)

x=0.00001
y=1.
u=0.0
h=0.001

while (x<15.5):
    yn=y
    y=y+(h*u)
    u=u+(h*((-(2./x)*u)-(yn)**5))
    x=x+h
    print(x,y)

WARNING: some intermediate output was truncated.
WARNING: 11 intermediate output messages were discarded.

 0.172767765981327
9.876009999999967 0.17275080680769264
9.877009999999967 0.17273385091489757
9.878009999999966 0.17271689830200515
9.879009999999965 0.17269994896807905
9.880009999999965 0.17268300291218333
9.881009999999964 0.17266606013338234
9.882009999999964 0.17264912063074087
9.883009999999963 0.17263218440332395
9.884009999999963 0.17261525145019707
9.885009999999962 0.172598321770426
9.886009999999962 0.17258139536307687
9.887009999999961 0.1725644722272162
9.88800999999996 0.17254755236191077
9.88900999999996 0.1725306357662278
9.89000999999996 0.1725137224392348
9.891009999999959 0.17249681237999967
9.892009999999958 0.17247990558759066
9.893009999999958 0.17246300206107631
9.894009999999957 0.17244610179952558
9.895009999999957 0.1724292048020077
9.896009999999956 0.17241231106759233
9.897009999999955 0.17239542059534943
9.898009999999955 0.1723785333843493
9.899009999999954 0.1723616494336626
9.900009999999954 0.17234476874236038
9.901009999999953 0.17232789130951395
9.902009999999953 0.17231101713419503
9.903009999999952 0.17229414621547567
9.904009999999952 0.17227727855242828
9.905009999999951 0.17226041414412557
9.90600999999995 0.17224355298964067
9.90700999999995 0.172226695088047
9.90800999999995 0.17220984043841833
9.909009999999949 0.1721929890398288
9.910009999999948 0.17217614089135286
9.911009999999948 0.17215929599206534
9.912009999999947 0.1721424543410414
9.913009999999947 0.17212561593735654
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9.923009999999941 0.1719574103010157
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9.92500999999994 0.17192380804954183
9.92600999999994 0.1719070117772929
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9.929009999999938 0.171856642363463
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9.935009999999934 0.17175599076625805
9.936009999999934 0.17173922679700312
9.937009999999933 0.1717224660530025
9.938009999999933 0.1717057085333403
9.939009999999932 0.17168895423710093
9.940009999999932 0.17167220316336915
9.941009999999931 0.17165545531123005
9.94200999999993 0.17163871067976907
9.94300999999993 0.17162196926807202
9.94400999999993 0.171605231075225
9.945009999999929 0.17158849610031446
9.946009999999928 0.17157176434242724
9.947009999999928 0.17155503580065046
9.948009999999927 0.1715383104740716
9.949009999999927 0.17152158836177847
9.950009999999926 0.17150486946285926
9.951009999999926 0.17148815377640245
9.952009999999925 0.17147144130149686
9.953009999999924 0.1714547320372317
9.954009999999924 0.17143802598269645
9.955009999999923 0.171421323136981
9.956009999999923 0.1714046234991755
9.957009999999922 0.17138792706837053
9.958009999999922 0.17137123384365693
9.959009999999921 0.1713545438241259
9.96000999999992 0.171337857008869
9.96100999999992 0.17132117339697808
9.96200999999992 0.1713044929875454
9.963009999999919 0.17128781577966348
9.964009999999918 0.17127114177242522
9.965009999999918 0.17125447096492386
9.966009999999917 0.17123780335625297
9.967009999999917 0.17122113894550642
9.968009999999916 0.1712044777317785
9.969009999999916 0.17118781971416372
9.970009999999915 0.17117116489175704
9.971009999999914 0.1711545132636537
9.972009999999914 0.17113786482894924
9.973009999999913 0.17112121958673965
9.974009999999913 0.17110457753612113
9.975009999999912 0.1710879386761903
9.976009999999912 0.17107130300604403
9.977009999999911 0.17105467052477963
9.97800999999991 0.1710380412314947
9.97900999999991 0.17102141512528712
9.98000999999991 0.1710047922052552
9.981009999999909 0.17098817247049752
9.982009999999908 0.170971555920113
9.983009999999908 0.17095494255320093
9.984009999999907 0.17093833236886088
9.985009999999907 0.17092172536619282
9.986009999999906 0.170905121544297
9.987009999999906 0.17088852090227402
9.988009999999905 0.1708719234392248
9.989009999999904 0.17085532915425064
9.990009999999904 0.1708387380464531
9.991009999999903 0.17082215011493415
9.992009999999903 0.17080556535879604
9.993009999999902 0.17078898377714138
9.994009999999902 0.1707724053690731
9.995009999999901 0.17075583013369444
9.9960099999999 0.170739258070109
9.9970099999999 0.17072268917742073
9.9980099999999 0.17070612345473388
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10.000009999999898 0.17067300151578313
10.001009999999898 0.17065644529772941
10.002009999999897 0.17063989224609746
10.003009999999897 0.17062334235999319
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10.009009999999893 0.17052410946942204
10.010009999999893 0.170507581717005
10.011009999999892 0.1704910571229749
10.012009999999892 0.1704745356864406
10.013009999999891 0.17045801740651134
10.01400999999989 0.17044150228229665
10.01500999999989 0.17042499031290642
10.01600999999989 0.17040848149745083
10.017009999999889 0.17039197583504045
10.018009999999888 0.17037547332478614
10.019009999999888 0.17035897396579908
10.020009999999887 0.17034247775719077
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10.023009999999886 0.1702930080247587
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10.036009999999878 0.1700789661472445
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10.039009999999877 0.17002964718632285
10.040009999999876 0.1700132137992143
10.041009999999876 0.1699967835439021
10.042009999999875 0.16998035641950499
10.043009999999875 0.16996393242514196
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10.14100999999982 0.16836942363671048
10.14200999999982 0.1683533052085234
10.14300999999982 0.16833718982389112
…0.15937541189325524
10.731009999999493 0.1593609489929154
10.732009999999493 0.15934648868553375
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10.838009999999434 0.15782826161774222
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WARNING: 11 intermediate output messages were discarded.
WARNING: some intermediate output was truncated.

#Solve the equation analytically for special cases n=0,1,5
#There is no solution (root) for n=5

#Solve for n=0:
from sympy.interactive import printing
printing.init_printing(use_latex=True)
from sympy import *
x = sp.Symbol('xi')
y = sp.Function('theta')(x)
x0=0
diffeq = Eq(y.diff(x,x)+(2/x)*y.diff(x)+y**0,0)
display(diffeq)

$\displaystyle \frac{d^{2}}{d \xi^{2}} \theta{\left(\xi \right)} + 1 + \frac{2 \frac{d}{d \xi} \theta{\left(\xi \right)}}{\xi} = 0$

sol = dsolve(diffeq,y)

display(sol)

$\displaystyle \theta{\left(\xi \right)} = C_{1} + \frac{C_{2}}{\xi} - \frac{\xi^{2}}{6}$

#Where C1=1 and C2=0 in accordance to inital conditions:
#theta(0)=1, theta'(0)=0

#When n=0:
x = symbols('xi')
eq1 = Eq(1-(1/6)*x**2,0)
sol = solve(eq1, dict=True)
print('The solution is:',sol[1])

The solution is: {xi: 2.44948974278318}

#When n=1:
x = symbols('xi')
eq2 = Eq(sp.sin(x)/x,0)
sol = solve(eq2, dict=True)
print('The solution is:',sol)

The solution is: [{xi: pi}]

#When n=5:
x = symbols('xi')
eq3 = Eq(1/(1+(x**2/3)),0)
sol = solve(eq3, dict=True)
print('The solution is:',sol)

The solution is: []

#Plot the differential equations

plt.plot(t, yz[:,0], label = 'n=0')
plt.plot(t, ya[:,0], label = 'n=1')
plt.plot(t, yb[:,0], label = 'n=2')
plt.plot(t, yc[:,0], label = 'n=3')
plt.plot(t, ye[:,0], label = 'n=4')
plt.plot(t, yp[:,0], label = 'n=5')
plt.legend()
plt.xlim(0,16)
plt.ylim(0,1.1)
plt.title('Solutions to the Lane-Emden Equation')
plt.xlabel('ξ')
plt.ylabel('θ')
plt.savefig('LEplot.pdf', bbox='tight')

/ext/anaconda-2019.03/lib/python3.7/site-packages/ipykernel/__main__.py:13: UserWarning: Creating legend with loc="best" can be slow with large amounts of data.
/ext/anaconda-2019.03/lib/python3.7/site-packages/ipykernel/__main__.py:13: UserWarning: Creating legend with loc="best" can be slow with large amounts of data.
/ext/anaconda-2019.03/lib/python3.7/site-packages/IPython/core/pylabtools.py:125: UserWarning: Creating legend with loc="best" can be slow with large amounts of data.
  fig.canvas.print_figure(bytes_io, **kw)

#Interpolate solutions of the differential equations
#Plot linear regression lines

x1 = 2.45
y1 = 0

x2 = 0
y2 = 1

a = (y2 - y1) / (x2 - x1)
b = y1 - a * x1
print('slope: ', a)
print('intercept: ', b)

slope:  -0.4081632653061224
intercept:  1.0

import scipy
from scipy import interpolate
from scipy.optimize import fsolve
x1 = np.linspace(0,2.45)
x2 = np.linspace(0,3.14)
x3 = np.linspace(0,4.35)
x4 = np.linspace(0,6.90)
x5 = np.linspace(0,15)
y = np.linspace(1,0)
f1 = interpolate.interp1d(x1, y, kind='linear')
f2 = interpolate.interp1d(x2, y, kind='linear')
f3 = interpolate.interp1d(x3, y, kind='linear')
f4 = interpolate.interp1d(x4, y, kind='linear')
f5 = interpolate.interp1d(x5, y, kind='linear')
plt.plot(x1,y)
plt.plot(x2,y)
plt.plot(x3,y)
plt.plot(x4,y)
plt.plot(x5,y)
plt.xlim(0,16)
plt.ylim(0,1.1)
plt.title('Linear Regression of the Lane-Emden Equation')
plt.xlabel('ξ')
plt.ylabel('θ')

Text(0, 0.5, 'θ')

root = scipy.optimize.fsolve(f1, 2)
print("xi =",root)

xi = [2.45]

#Load SSM data from text file in cocalc
datafile = np.genfromtxt('bs05_agsop.dat.txt', skip_header=23, skip_footer=2)

#Split SSM data by columns
mass = datafile[:,0]
radius = datafile[:,1]
temperature = datafile[:,2]
density = datafile[:,3]
pressure = datafile[:,4]

#Convert to logarithmic values
logpressure = np.log(pressure)
logdensity = np.log(density)
logtemp = np.log(temperature)

#Visualise all four plots, sharing the same horizontal axis
fig = plt.figure()
fig, (ax1,ax2,ax3,ax4) = plt.subplots(4, sharex=True,gridspec_kw={'hspace': 0.2})
fig.suptitle('Standard Solar Model')
ax1.plot(radius, mass)
ax1.set_ylabel('Mass (Msun)')
ax2.plot(radius, logdensity)
ax2.set_ylabel('Log rho')
ax3.plot(radius, logpressure)
ax3.set_ylabel('Log P')
ax4.plot(radius, logtemp)
ax4.set_xlabel('Radius (Rsun)')
ax4.set_ylabel('Log T')
plt.gca().minorticks_on()
plt.savefig('SSM.pdf')

<Figure size 432x288 with 0 Axes>

import nbconvert

help(nbconvert)

Help on package nbconvert:

NAME
    nbconvert - Utilities for converting notebooks to and from different formats.

PACKAGE CONTENTS
    __main__
    _version
    exporters (package)
    filters (package)
    nbconvertapp
    postprocessors (package)
    preprocessors (package)
    resources (package)
    tests (package)
    utils (package)
    writers (package)

SUBMODULES
    asciidoc
    base
    exporter
    html
    latex
    markdown
    notebook
    pdf
    python
    rst
    script
    slides
    templateexporter

DATA
    version_info = (5, 6, 1)

VERSION
    5.6.1

FILE
    /ext/anaconda-2019.03/lib/python3.7/site-packages/nbconvert/__init__.py