{"text_format":"1","workbook":"xlsx","n_sheets":6,"sheets":["RK Instructions","Set Up","RungeKutta","Summary","RK Plot","revision"],"not_rendered":{"charts":2,"images":2,"macros":1,"external_links":0,"defined_names":26,"comments":0,"pivot_tables":0},"unopened":[{"sheet":"RK Plot","index":4,"reason":"not_a_worksheet"}],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"RK Instructions","index":0,"n_rows":0,"n_cols":0,"rows_kept":0,"cut":false,"hidden":false,"merged":[]}

{"sheet":"Set Up","index":1,"n_rows":22,"n_cols":3,"rows_kept":22,"cut":false,"hidden":false,"merged":[]}
["Enter Independent Variable Information"]
["Name","Initial Value","Final  Value"]
["tau",0,0.5]
["Num of Steps",null,100]
["Step Size (calc'd)",null,{"$f":"(finalindep-indep)/numsteps","v":5.0000000000000001E-3}]
["Number of Points in Summary",null,20]
["Enter Constants, Initial Values, Formulas, and Differential Equations"]
["Name","Input","Display Formula"]
["k.1",55.5]
["k.2",30.2]
["c.A",1.4999999999999999E-2]
["c.B",2.1000000000000001E-2]
["c.C",0]
["c.D",0]
["c.E",0]
["r.1",{"$f":"B9*B11*SQRT(B12)","v":0.12064071141202708},"=B9*B11*SQRT(B12)"]
["r.2",{"$f":"B10*B13*SQRT(B12)","v":0},"=B10*B13*SQRT(B12)"]
["d(c.A)/d(tau)",{"$f":"-B16","v":-0.12064071141202708},"=-B16"]
["d(c.B)/d(tau)",{"$f":"-B16-B17","v":-0.12064071141202708},"=-B16-B17"]
["d(c.C)/d(tau)",{"$f":"B16-B17","v":0.12064071141202708},"=B16-B17"]
["d(c.D)/d(tau)",{"$f":"B16+B17","v":0.12064071141202708},"=B16+B17"]
["d(c.E)/d(tau)",{"$f":"B17","v":0},"=B17"]

{"sheet":"RungeKutta","index":2,"n_rows":103,"n_cols":8,"rows_kept":103,"cut":false,"hidden":false,"merged":[]}
["Runge Kutta Output"]
["tau","c.A","c.B","c.C","c.D","c.E","r.1","r.2"]
[0,1.4999999999999999E-2,2.1000000000000001E-2,0,0,0,0.12064071141202708,0]
[5.0000000000000001E-3,1.4412900201308754E-2,2.040652654051462E-2,5.8072613789711121E-4,5.934734594853795E-4,6.373660794134145E-6,0.11426811745774719,2.5061019790507217E-3]
[0.01,1.3856635807303888E-2,1.9832003682106181E-2,1.1187320674984048E-3,1.1679963178938182E-3,2.4632125197706695E-5,0.1083004808456827,4.7586172308089732E-3]
[1.4999999999999999E-2,1.3329262527823528E-2,1.9275692820537402E-2,1.617167764890346E-3,1.7243071794625984E-3,5.3569707286126112E-5,0.10270711624981722,6.7812189117889233E-3]
[0.02,1.2828977790436596E-2,1.8736888445333284E-2,2.0789328644600915E-3,2.263111554666717E-3,9.2089345103312469E-5,9.7460880143012171E-2,8.5945891834737029E-3]
[2.5000000000000001E-2,1.2354109117927529E-2,1.8214916587293524E-2,2.506698351438465E-3,2.7850834127064786E-3,1.391925306340066E-4,9.2536844504611665E-2,1.0217498713583024E-2]
[3.0000000000000002E-2,1.190310355829231E-2,1.7709133330760914E-2,2.9029262141762953E-3,3.2908666692390877E-3,1.9397022753139606E-4,8.7912097088483604E-2,1.1666993512464573E-2]
[3.5000000000000003E-2,1.1474518063534772E-2,1.7218923389927893E-2,3.2698872628583496E-3,3.7810766100721096E-3,2.555946736068799E-4,8.356556128929539E-2,1.2958562358137476E-2]
[0.04,1.1067010724539325E-2,1.6743698748064591E-2,3.6096772989859418E-3,4.2563012519354123E-3,3.2331197647473517E-4,7.9477833538844522E-2,1.4106286948423502E-2]
[4.4999999999999998E-2,1.0679332779044903E-2,1.6282897358242222E-2,3.9242318001524159E-3,4.7171026417577814E-3,3.9643542080268306E-4,7.5631036396585014E-2,1.5122976670054852E-2]
[4.9999999999999996E-2,1.0310321318402194E-2,1.5835981903890017E-2,4.2153392670856264E-3,5.1640180961099865E-3,4.7433941451218017E-4,7.2008685703293537E-2,1.6020289656393828E-2]
[5.4999999999999993E-2,9.9588926264909516E-3,1.5402438617349632E-2,4.4846533643677295E-3,5.5975613826503706E-3,5.564540091413209E-4,6.8595570347505602E-2,1.6808841613593594E-2]
[5.9999999999999991E-2,9.6240360910194549E-3,1.4981776154467217E-2,4.7337039724283068E-3,6.0182238455327858E-3,6.422599365522399E-4,6.5377643353895662E-2,1.7498303726330083E-2]
[6.4999999999999988E-2,9.304808633523079E-3,1.4573524523181121E-2,4.9639072561349605E-3,6.4264754768188831E-3,7.3128411034196171E-4,6.2341923143766957E-2,1.809749080573821E-2]
[6.9999999999999993E-2,9.0003296098103269E-3,1.4177234064015005E-2,5.1765748443943511E-3,6.8227659359849976E-3,8.2309554579532401E-4,5.9476403942516493E-2,1.8614440711365532E-2]
[7.4999999999999997E-2,8.7097761374491951E-3,1.3792474480365531E-2,5.3729222054671377E-3,7.2075255196344728E-3,9.1730165708366811E-4,5.676997441932468E-2,1.9056485963614037E-2]
[0.08,8.4323788112113766E-3,1.3418833916475434E-2,5.5540762940526778E-3,7.5811660835245704E-3,1.0135448947359467E-3,5.4212343742113589E-2,1.9430318361362126E-2]
[8.5000000000000006E-2,8.1674177712556997E-3,1.3055918081002401E-2,5.7210825384909972E-3,7.9440819189976041E-3,1.1114996902533037E-3,5.1793974317529606E-2,1.9742047329574814E-2]
[9.0000000000000011E-2,7.9142190922875111E-3,1.2703349414127723E-2,5.8749112295526972E-3,8.2966505858722821E-3,1.2108696781597931E-3,4.9506020562658758E-2,1.999725264226597E-2]
[9.5000000000000015E-2,7.6721514650230672E-3,1.2360766296193393E-2,6.0164633661472554E-3,8.6392337038066116E-3,1.3113851688296787E-3,4.7340273123530112E-2,2.0201032095902249E-2]
[0.10000000000000002,7.4406231440580957E-3,1.2027822295909381E-2,6.1465760077931871E-3,8.9721777040906239E-3,1.4128008481487191E-3,4.5289108016218549E-2,2.0358044646124888E-2]
[0.10500000000000002,7.2190791387229713E-3,1.170418545623216E-2,6.2660271787862145E-3,9.2958145437678445E-3,1.514893682490816E-3,4.3345440220412083E-2,2.0472549465544044E-2]
[0.11000000000000003,7.0069986257351542E-3,1.1389537616079472E-2,6.3755403646091587E-3,9.6104623839205327E-3,1.6174610096556883E-3,4.1502681303437712E-2,2.0548441331482335E-2]
[0.11500000000000003,6.8038925644603775E-3,1.1083573766113224E-2,6.4757886371924643E-3,9.9164262338867805E-3,1.720318798347159E-3,3.9754700695632215E-2,2.0589282709168891E-2]
[0.12000000000000004,6.6093014973922955E-3,1.0786001436891249E-2,6.5673984421066546E-3,1.0213998563108756E-2,1.8233000605010516E-3,3.8095790276198277E-2,2.0598332857361026E-2]
[0.12500000000000003,6.4227935200777105E-3,1.0496540117758321E-2,6.6509530776028957E-3,1.0503459882241684E-2,1.926253402319395E-3,3.6520631962833783E-2,2.0578574249129546E-2]
[0.13000000000000003,6.2439624061704528E-3,1.0214920704916679E-2,6.7269958925757697E-3,1.0785079295083325E-2,2.0290417012537782E-3,3.5024268028927732E-2,2.0532736570084135E-2]
[0.13500000000000004,6.0724258746086097E-3,9.9408849771855639E-3,6.7960332279683412E-3,1.1059115022814441E-2,2.1315408974230498E-3,3.3602073899390843E-2,2.0463318529200029E-2]
[0.14000000000000004,5.9078239870924091E-3,9.6741850980275341E-3,6.8585371238427117E-3,1.1325814901972471E-2,2.2336388890648799E-3,3.2249733200594301E-2,2.0372607693248966E-2]
[0.14500000000000005,5.7498176651070883E-3,9.4145831424861954E-3,6.9149478122720147E-3,1.1585416857513809E-2,2.3352345226208978E-3,3.0963214861745705E-2,2.0262698534298792E-2]
[0.15000000000000005,5.5980873166986307E-3,9.1618506477450963E-3,6.9656760143478316E-3,1.1838149352254908E-2,2.4362366689535384E-3,2.9738752084618732E-2,2.013550886052709E-2]
[0.15500000000000005,5.4523315640809854E-3,8.9157681860806987E-3,7.0111050579187246E-3,1.2084231813919306E-2,2.5365633780002907E-3,2.8572823016121886E-2,1.9992794783433368E-2]
[0.16000000000000006,5.3122660639408808E-3,8.6761249590434405E-3,7.0515928311616757E-3,1.2323875040956564E-2,2.6361411048974443E-3,2.7462132973961956E-2,1.9836164359198889E-2]
[0.16500000000000006,5.1776224130190233E-3,8.4427184117597271E-3,7.0874735857216773E-3,1.2557281588240278E-2,2.7349040012593006E-3,2.6403598089824839E-2,1.9667090028229656E-2]
[0.17000000000000007,5.0481471321920076E-3,8.2153538663042504E-3,7.1190596019202312E-3,1.2784646133695754E-2,2.8327932658877616E-3,2.5394330247232422E-2,1.9486919964645549E-2]
[0.17500000000000007,4.9236007228643311E-3,7.9938441731462508E-3,7.1466427274175844E-3,1.3006155826853754E-2,2.9297565497180848E-3,2.4431623202693348E-2,1.929688843648783E-2]
[0.18000000000000008,4.8037567900106005E-3,7.7780093797252129E-3,7.1704957997040077E-3,1.3221990620274793E-2,3.0257474102853925E-3,2.3512939789082553E-2,1.909812526756768E-2]
[0.18500000000000008,4.6884012266897252E-3,7.5676764152610449E-3,7.1908739618815904E-3,1.3432323584738962E-2,3.1207248114286856E-3,2.2635900109480595E-2,1.8891664483044773E-2]
[0.19000000000000009,4.5773314552904308E-3,7.3626787909510443E-3,7.2080158803701794E-3,1.3637321209048961E-2,3.2146526643393914E-3,2.1798270638086682E-2,1.8678452212897521E-2]
[0.19500000000000009,4.470355721165142E-3,7.1628563147509575E-3,7.2221448724206702E-3,1.3837143685249048E-2,3.3074994064141894E-3,2.0997954152383112E-2,1.8459353920327239E-2]
[0.20000000000000009,4.3672924346710466E-3,6.9680548199802473E-3,7.2334699506381508E-3,1.4031945180019758E-2,3.3992376146908042E-3,2.0232980427559503E-2,1.8235161015740426E-2]
[0.2050000000000001,4.2679695579664282E-3,6.778125907032387E-3,7.2421867910995264E-3,1.4221874092967619E-2,3.4898436509340465E-3,1.9501497630377736E-2,1.8006596911199872E-2]
[0.2100000000000001,4.1722240332102718E-3,6.5929266975095907E-3,7.248478631089043E-3,1.4407073302490414E-2,3.5792973357006859E-3,1.8801764355240235E-2,1.7774322565057926E-2]
[0.21500000000000011,4.0799012490865131E-3,6.4123196001380665E-3,7.2525171019650362E-3,1.4587680399861939E-2,3.6675816489484519E-3,1.8132142250275488E-2,1.7538941561823104E-2]
[0.22000000000000011,3.9908545428236101E-3,6.2361720878545759E-3,7.2544630022073524E-3,1.4763827912145431E-2,3.7546824549690391E-3,1.7491089185828926E-2,1.7301004768110698E-2]
[0.22500000000000012,3.9049447351076536E-3,6.0643564854880264E-3,7.2544670152727159E-3,1.493564351451198E-2,3.8405882496196321E-3,1.6877152921892097E-2,1.7061014601740351E-2]
[0.23000000000000012,3.8220396954950289E-3,5.896749767490949E-3,7.252670376500887E-3,1.5103250232509058E-2,3.9252899280040852E-3,1.6288965234762032E-2,1.681942894762695E-2]
[0.23500000000000013,3.7420139361205155E-3,5.7332333652052219E-3,7.2492054929641867E-3,1.5266766634794786E-2,4.0087805709152994E-3,1.5725236466633215E-2,1.6576664751026197E-2]
[0.24000000000000013,3.6647482316703468E-3,5.5736929831742927E-3,7.2441965198335951E-3,1.5426307016825715E-2,4.0910552484960602E-3,1.5184750464921511E-2,1.6333101315910155E-2]
[0.24500000000000013,3.5901292637486077E-3,5.4180184240405294E-3,7.2377598965433099E-3,1.5581981575959478E-2,4.1721108397080849E-3,1.4666359880933285E-2,1.6089083333729378E-2]
[0.25000000000000011,3.5180492879107956E-3,5.2661034215912813E-3,7.230004845769686E-3,1.5733896578408728E-2,4.2519458663195205E-3,1.4168981800051297E-2,1.5844923665540665E-2]
[0.25500000000000012,3.4484058217715951E-3,5.1178454815407996E-3,7.2210338379976053E-3,1.5882154518459209E-2,4.3305603402308017E-3,1.3691593677936443E-2,1.5600905898418328E-2]
[0.26000000000000012,3.3811013527160596E-3,4.9731457296574508E-3,7.2109430242253275E-3,1.6026854270342557E-2,4.407955623058615E-3,1.3233229559363348E-2,1.5357286695200565E-2]
[0.26500000000000012,3.3160430638553904E-3,4.8319087668666945E-3,7.1998226391559094E-3,1.6168091233133314E-2,4.4841342969887017E-3,1.2792976558237803E-2,1.5114297954931752E-2]
[0.27000000000000013,3.2531425769713176E-3,4.6940425309802015E-3,7.1877573770375621E-3,1.6305957469019808E-2,4.5591000459911223E-3,1.2369971579103542E-2,1.4872148799828757E-2]
[0.27500000000000013,3.1923157112874606E-3,4.5594581647202381E-3,7.1748267421453136E-3,1.6440541835279772E-2,4.6328575465672279E-3,1.1963398262050258E-2,1.463102740320925E-2]
[0.28000000000000014,3.1334822569927911E-3,4.4280698897262195E-3,7.161105375740634E-3,1.6571930110273789E-2,4.7054123672665765E-3,1.1572484134399093E-2,1.439110267155831E-2]
[0.28500000000000014,3.0765657625220363E-3,4.2997948862470506E-3,7.1466633612029748E-3,1.6700205113752958E-2,4.7767708762749906E-3,1.1196497953879116E-2,1.4152525792764312E-2]
[0.29000000000000015,3.0214933346711958E-3,4.1745531782387225E-3,7.1315665088963276E-3,1.6825446821761285E-2,4.8469401564324782E-3,1.0834747229229939E-2,1.3915431661514127E-2]
[0.29500000000000015,2.9681954506938458E-3,4.052267523601564E-3,7.1158766222138691E-3,1.6947732476398445E-2,4.9159279270922871E-3,1.0486575905282583E-2,1.3679940191891924E-2]
[0.30000000000000016,2.9166057815860407E-3,3.9328633093056505E-3,7.0996517461335657E-3,1.7067136690694359E-2,4.9837424722803956E-3,1.0151362200592614E-2,1.3446157526365542E-2]
[0.30500000000000016,2.8666610258248921E-3,3.8162684511662166E-3,7.0829463995164292E-3,1.7183731548833792E-2,5.0503925746586808E-3,9.8285165866344882E-3,1.3214177149561438E-2]
[0.31000000000000016,2.8183007528786819E-3,3.7024132980434852E-3,7.0658117922861186E-3,1.7297586701956524E-2,5.115887454835202E-3,9.5174798984228211E-3,1.2984080914516806E-2]
[0.31500000000000017,2.7714672558550508E-3,3.5912305402532284E-3,7.0482960285431235E-3,1.740876945974678E-2,5.1802367156018282E-3,9.2177215672108164E-3,1.2755939988448119E-2]
[0.32000000000000017,2.7261054126987323E-3,3.4826551219855981E-3,7.0304442965881293E-3,1.7517344878014412E-2,5.2434502907131404E-3,8.9287379666357861E-3,1.2529815724484035E-2]
[0.32500000000000018,2.6821625553917713E-3,3.3766241575403674E-3,7.0122990467568211E-3,1.7623375842459644E-2,5.3055383978514101E-3,8.6500508643416133E-3,1.2305760465271109E-2]
[0.33000000000000018,2.6395883466474881E-3,3.2730768511967509E-3,6.9939001579017715E-3,1.772692314880326E-2,5.366511495450743E-3,8.3812059717139117E-3,1.2083818283868893E-2]
[0.33500000000000019,2.5983346636248624E-3,3.1719544205454355E-3,6.9752850932957074E-3,1.7828045579454575E-2,5.4263802430794322E-3,8.1217715849198346E-3,1.1864025666902039E-2]
[0.34000000000000019,2.5583554882227491E-3,3.0732000231194034E-3,6.9564890466739023E-3,1.7926799976880608E-2,5.485155465103351E-3,7.8713373109557602E-3,1.1646412144527052E-2]
[0.3450000000000002,2.5196068035436359E-3,2.9767586861685767E-3,6.9375450790813021E-3,1.8023241313831435E-2,5.5428481173750649E-3,7.6295128728758746E-3,1.1431000871396941E-2]
[0.3500000000000002,2.4820464961446938E-3,2.8825772394313136E-3,6.9184842471419232E-3,1.8117422760568699E-2,5.5994692567133859E-3,7.3959269888069154E-3,1.1217809162464771E-2]
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[0.36000000000000021,2.4103315278805426E-3,2.7007899644918066E-3,6.8801269087307186E-3,1.8299210035508207E-2,5.7095415633887422E-3,6.9520744815531514E-3,1.0798127425139702E-2]
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{"sheet":"Summary","index":3,"n_rows":23,"n_cols":8,"rows_kept":23,"cut":false,"hidden":false,"merged":[]}
["Summary of Output"]
["tau","c.A","c.B","c.C","c.D","c.E","r.1","r.2"]
[0,1.4999999999999999E-2,2.1000000000000001E-2,0,0,0,0.12064071141202708,0]
[2.5000000000000001E-2,1.2354109117927529E-2,1.8214916587293524E-2,2.506698351438465E-3,2.7850834127064786E-3,1.391925306340066E-4,9.2536844504611665E-2,1.0217498713583024E-2]
[4.9999999999999996E-2,1.0310321318402194E-2,1.5835981903890017E-2,4.2153392670856264E-3,5.1640180961099865E-3,4.7433941451218017E-4,7.2008685703293537E-2,1.6020289656393828E-2]
[7.4999999999999997E-2,8.7097761374491951E-3,1.3792474480365531E-2,5.3729222054671377E-3,7.2075255196344728E-3,9.1730165708366811E-4,5.676997441932468E-2,1.9056485963614037E-2]
[0.10000000000000002,7.4406231440580957E-3,1.2027822295909381E-2,6.1465760077931871E-3,8.9721777040906239E-3,1.4128008481487191E-3,4.5289108016218549E-2,2.0358044646124888E-2]
[0.12500000000000003,6.4227935200777105E-3,1.0496540117758321E-2,6.6509530776028957E-3,1.0503459882241684E-2,1.926253402319395E-3,3.6520631962833783E-2,2.0578574249129546E-2]
[0.15000000000000005,5.5980873166986307E-3,9.1618506477450963E-3,6.9656760143478316E-3,1.1838149352254908E-2,2.4362366689535384E-3,2.9738752084618732E-2,2.013550886052709E-2]
[0.17500000000000007,4.9236007228643311E-3,7.9938441731462508E-3,7.1466427274175844E-3,1.3006155826853754E-2,2.9297565497180848E-3,2.4431623202693348E-2,1.929688843648783E-2]
[0.20000000000000009,4.3672924346710466E-3,6.9680548199802473E-3,7.2334699506381508E-3,1.4031945180019758E-2,3.3992376146908042E-3,2.0232980427559503E-2,1.8235161015740426E-2]
[0.22500000000000012,3.9049447351076536E-3,6.0643564854880264E-3,7.2544670152727159E-3,1.493564351451198E-2,3.8405882496196321E-3,1.6877152921892097E-2,1.7061014601740351E-2]
[0.25000000000000011,3.5180492879107956E-3,5.2661034215912813E-3,7.230004845769686E-3,1.5733896578408728E-2,4.2519458663195205E-3,1.4168981800051297E-2,1.5844923665540665E-2]
[0.27500000000000013,3.1923157112874606E-3,4.5594581647202381E-3,7.1748267421453136E-3,1.6440541835279772E-2,4.6328575465672279E-3,1.1963398262050258E-2,1.463102740320925E-2]
[0.30000000000000016,2.9166057815860407E-3,3.9328633093056505E-3,7.0996517461335657E-3,1.7067136690694359E-2,4.9837424722803956E-3,1.0151362200592614E-2,1.3446157526365542E-2]
[0.32500000000000018,2.6821625553917713E-3,3.3766241575403674E-3,7.0122990467568211E-3,1.7623375842459644E-2,5.3055383978514101E-3,8.6500508643416133E-3,1.2305760465271109E-2]
[0.3500000000000002,2.4820464961446938E-3,2.8825772394313136E-3,6.9184842471419232E-3,1.8117422760568699E-2,5.5994692567133859E-3,7.3959269888069154E-3,1.1217809162464771E-2]
[0.37500000000000022,2.310718674945264E-3,2.4438256876773345E-3,6.8223883377868041E-3,1.855617431232268E-2,5.8668929872679352E-3,6.339783328934959E-3,1.0185400470941476E-2]
[0.40000000000000024,2.1637296719103374E-3,2.0545269559293792E-3,6.727067612108702E-3,1.8945473044070637E-2,6.109202715980964E-3,5.4431604058875769E-3,9.2084855962586299E-3]
[0.42500000000000027,2.0374852745864576E-3,1.7097217603902059E-3,6.6347512112172879E-3,1.9290278239609809E-2,6.3277635141962578E-3,4.6757293542047411E-3,8.2850242628531197E-3]
[0.45000000000000029,1.9290685559425851E-3,1.4051956859522867E-3,6.5470585740671139E-3,1.9594804314047729E-2,6.5238728699903046E-3,4.0133603497412382E-3,7.4117532619182433E-3]
[0.47500000000000031,1.8361037585718411E-3,1.1373668401448071E-3,6.4651593230011231E-3,1.986263315985521E-2,6.6987369184270396E-3,3.4366829378951421E-3,6.5846955181074916E-3]
[0.50000000000000033,1.7566514824619616E-3,9.0319441780876563E-4,6.3898914528848416E-3,2.0096805582191252E-2,6.8534570646532006E-3,3.0263989178849274E-3,5.9534141299760644E-3]

{"sheet":"revision","index":5,"n_rows":4,"n_cols":3,"rows_kept":4,"cut":false,"hidden":false,"merged":[]}
[]
["ver"]
["2.0","2012-12-31","revised to eliminate labels"]
[null,null,"extensive revisions to macro for clarity"]