{"text_format":"1","workbook":"xlsx","n_sheets":1,"sheets":["RK4"],"not_rendered":{"charts":1,"images":0,"macros":1,"external_links":0,"defined_names":2,"comments":0,"pivot_tables":0},"unopened":[],"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":"RK4","index":0,"n_rows":28,"n_cols":11,"rows_kept":28,"cut":false,"hidden":false,"merged":[]}
["2020-03-07",null,"Want to solve  y' = f(x,y)  with y(x0) = y0",null,null,null,null,null,"f = x + y",null,"Mathematica"]
[null,"Numerical Solutions of Differential Equations - J. Mahaffy, San Diego State University",null,null,null,null,null,null,null,"https://jmahaffy.sdsu.edu/courses.html"]
[null,null,"Runge-Kutta Method",null,null,null,null,"c =",3,null,"sol = DSolve[y'[t] == t + y[t], y, t]"]
[null,"x0 =",0,"y0 =",2,"h=",0.1,null,null,null,"y = c*EXP(x)-1-x"]
[null,null,null,null,"f(x+h/2,y+k1h/2)",null,"f(x+h,y+k3h)"]
[null,null,null,"f(x,y)",null,"f(x+h/2,y+k2h/2)",null,"y(n+1) = y(n)+(h/6)*(k1 + 2*k2 + 2*k3 + k4)"]
["n","x(n)","y(n)","k1","k2","k3","k4",null,"Analytic"]
[0,{"$f":"$C$4","v":0},{"$f":"$E$4","v":2},{"$f":"f(B8,C8)","v":2},{"$f":"f(B8 + h/2, C8 + h*D8/2)","v":2.15},{"$f":"f(B8 + h/2, C8 + h*E8/2)","v":2.1574999999999998},{"$f":"f(B8 + h, C8 + h*F8)","v":2.31575},{"$f":"C8+(h/6)*(D8 + 2*E8 + 2*F8 +G8 )","v":2.2155125},{"$f":"$I$3*EXP(B8)-1-B8","v":2}]
[{"$f":"A8+1","v":1},{"$f":"B8+$G$4","v":0.1},{"$f":"H8","v":2.2155125},{"$f":"f(B9,C9)","v":2.3155125000000001},{"$f":"f(B9 + h/2, C9 + h*D9/2)","v":2.4812881249999998},{"$f":"f(B9 + h/2, C9 + h*E9/2)","v":2.4895769062499999},{"$f":"f(B9 + h, C9 + h*F9)","v":2.6644701906250003},{"$f":"C9+(h/6)*(D9 + 2*E9 + 2*F9 +G9 )","v":2.4642077125520832},{"$f":"$I$3*EXP(B9)-1-B9","v":2.215512754226943}]
[{"$f":"A9+1","v":2},0.2,2.4642077125520832,2.6642077125520833,2.8474180981796873,2.8565786174610674,3.0498655742981899,2.7495754911876129,2.4642082744805096]
[3,0.30000000000000004,2.7495754911876129,3.0495754911876132,3.2520542657469935,3.2621782044749628,3.4757933116351092,3.0754727202420566,2.7495764227280093]
[4,0.4,3.0754727202420566,3.4754727202420566,3.6992463562541595,3.710435038054765,3.9465162240475333,3.4461619157905141,3.0754740929238111]
[5,0.5,3.4461619157905141,3.9461619157905141,4.1934700115800396,4.2058354163695162,4.4667454574274652,3.8663538862757991,3.4461638121003846]
[6,0.6,3.8663538862757991,4.4663538862757992,4.7396715805895893,4.7533374653052789,5.0416876328063269,4.3412548797903305,3.8663564011715263]
[7,0.7,4.3412548797903305,5.0412548797903307,5.3433176237798472,5.3584207609793228,5.6770969558882625,4.8766186898769464,4.3412581224114293]
[8,0.79999999999999993,4.8766186898769464,5.6766186898769462,6.0104496243707937,6.0271411710954856,6.3793328069864952,5.4788042413402129,4.8766227854774025]
[9,0.89999999999999991,5.4788042413402129,6.3788042413402124,6.7477444534072237,6.766191464010574,7.1554233877412701,6.1548392324054975,5.4788093334708474]
[10,0.99999999999999989,6.1548392324054975,7.1548392324054975,7.5625811940257721,7.582968292106786,8.0131360616161764,6.9124904701769436,6.1548454853771357]
[11,1.0999999999999999,6.9124904701769436,8.0124904701769442,8.4631149936857906,8.4856462198612324,8.9610550921630665,7.7603416033341777,6.9124980718392983]
[12,1.2,7.7603416033341777,8.960341603334177,9.4583586835008866,9.4832595375092215,10.008667557085101,8.7078790300415019,7.7603507682096415]
[13,1.3,8.7078790300415019,10.007879030041503,10.558272981543576,10.585792679118681,11.166458297953371,9.7655868408634916,8.7078900028577326]
[14,1.4000000000000001,9.7655868408634916,11.165586840863492,11.773866182906666,11.804280150008825,12.446014855864375,10.945051746906138,9.7655999005340259]
[15,1.5000000000000002,10.945051746906138,12.445051746906138,13.117304334251445,13.150916963618711,13.86014344326801,12.259079043338046,10.945067211014196]
[16,1.6000000000000003,12.259079043338046,13.859079043338046,14.602032995504949,14.639180693113294,15.422997112649377,13.721820768891778,12.25909727318535]
[17,1.7000000000000004,13.721820768891778,15.421820768891779,16.242911807336366,16.283966359258596,17.150217404817639,15.348917344006766,13.721842175181605]
[18,1.8000000000000005,15.348917344006766,17.148917344006765,18.056363211207106,18.101735504567124,19.05909089446348,17.157654105173744,15.348942393238847]
[19,1.9000000000000006,17.157654105173744,19.057654105173746,20.060536810432431,20.110680945695364,21.168722199743282,19.16713430212662,17.157683326837816]
[20,2.0000000000000004,19.16713430212662,21.16713430212662,22.275491017232952,22.330908852988269,23.50022518742545,21.398470289293194,19.16716829679196]