The Euler Method for approximation of 1st order ODEs
y'=f(t,y) Solution of equation y'=t-3y from t=0 to t=1, with initial condition y(0)=1, step size h=.001
f(t,y): t -3y initial time t_0: 0 initial value y_0: 1
solution y(t): (2/3)e^(-3t)-t/3+1/3
timestep h: 0.001 recursion: y_{n+1}=y_n+h*f(t_n,y_n) t_{n+1}=t_n+h
n t_n y_n y(t_n) f(t_n,y_n) y(t_n) y_n-y(t_n)
0 0 1 1 -3 1 0
1 0.001 0.997 0.99767 -2.99 0.99767 -0.00067
2 0.002 0.99401 0.995345 -2.98003 0.995345 -0.00134
3 0.003 0.99103 0.993027 -2.97009 0.993027 -0.002
4 0.004 0.98806 0.990714 -2.96018 0.990714 -0.00265
5 0.005 0.9851 0.988408 -2.950299 0.988408 -0.00331
6 0.006 0.982149 0.986107 -2.940448 0.986107 -0.00396
7 0.007 0.979209 0.983813 -2.930627 0.983813 -0.0046
8 0.008 0.976278 0.981524 -2.920835 0.981524 -0.00525
9 0.009 0.973357 0.979241 -2.911072 0.979241 -0.00588
10 0.01 0.970446 0.976964 -2.901339 0.976964 -0.00652
11 0.011 0.967545 0.974692 -2.891635 0.974692 -0.00715
12 0.012 0.964653 0.972427 -2.88196 0.972427 -0.00777
13 0.013 0.961771 0.970167 -2.872314 0.970167 -0.0084
14 0.014 0.958899 0.967913 -2.862698 0.967913 -0.00901
15 0.015 0.956036 0.965665 -2.853109 0.965665 -0.00963
16 0.016 0.953183 0.963423 -2.84355 0.963423 -0.01024
17 0.017 0.95034 0.961186 -2.834019 0.961186 -0.01085
18 0.018 0.947506 0.958955 -2.824517 0.958955 -0.01145
19 0.019 0.944681 0.956729 -2.815044 0.956729 -0.01205
20 0.02 0.941866 0.95451 -2.805599 0.95451 -0.01264
21 0.021 0.939061 0.952296 -2.796182 0.952296 -0.01324
22 0.022 0.936264 0.950087 -2.786793 0.950087 -0.01382
23 0.023 0.933478 0.947884 -2.777433 0.947884 -0.01441
24 0.024 0.9307 0.945687 -2.768101 0.945687 -0.01499
25 0.025 0.927932 0.943496 -2.758796 0.943496 -0.01556
26 0.026 0.925173 0.94131 -2.74952 0.94131 -0.01614
27 0.027 0.922424 0.939129 -2.740271 0.939129 -0.01671
28 0.028 0.919684 0.936954 -2.731051 0.936954 -0.01727
29 0.029 0.916952 0.934785 -2.721857 0.934785 -0.01783
30 0.03 0.914231 0.932621 -2.712692 0.932621 -0.01839
31 0.031 0.911518 0.930462 -2.703554 0.930462 -0.01894
32 0.032 0.908814 0.928309 -2.694443 0.928309 -0.01949
33 0.033 0.90612 0.926162 -2.68536 0.926162 -0.02004
34 0.034 0.903435 0.92402 -2.676304 0.92402 -0.02059
35 0.035 0.900758 0.921883 -2.667275 0.921883 -0.02112
36 0.036 0.898091 0.919752 -2.658273 0.919752 -0.02166
37 0.037 0.895433 0.917626 -2.649298 0.917626 -0.02219
38 0.038 0.892783 0.915505 -2.64035 0.915505 -0.02272
39 0.039 0.890143 0.91339 -2.631429 0.91339 -0.02325
40 0.04 0.887512 0.91128 -2.622535 0.91128 -0.02377
41 0.041 0.884889 0.909176 -2.613667 0.909176 -0.02429
42 0.042 0.882275 0.907077 -2.604826 0.907077 -0.0248
43 0.043 0.879671 0.904983 -2.596012 0.904983 -0.02531
44 0.044 0.877075 0.902894 -2.587224 0.902894 -0.02582
45 0.045 0.874487 0.900811 -2.578462 0.900811 -0.02632
46 0.046 0.871909 0.898732 -2.569727 0.898732 -0.02682
47 0.047 0.869339 0.89666 -2.561018 0.89666 -0.02732
48 0.048 0.866778 0.894592 -2.552335 0.894592 -0.02781
49 0.049 0.864226 0.892529 -2.543678 0.892529 -0.0283
50 0.05 0.861682 0.890472 -2.535047 0.890472 -0.02879
51 0.051 0.859147 0.88842 -2.526441 0.88842 -0.02927
52 0.052 0.856621 0.886373 -2.517862 0.886373 -0.02975
53 0.053 0.854103 0.884331 -2.509308 0.884331 -0.03023
54 0.054 0.851594 0.882294 -2.500781 0.882294 -0.0307
55 0.055 0.849093 0.880262 -2.492278 0.880262 -0.03117
56 0.056 0.8466 0.878236 -2.483801 0.878236 -0.03164
57 0.057 0.844117 0.876214 -2.47535 0.876214 -0.0321
58 0.058 0.841641 0.874198 -2.466924 0.874198 -0.03256
59 0.059 0.839174 0.872187 -2.458523 0.872187 -0.03301
60 0.06 0.836716 0.87018 -2.450148 0.87018 -0.03346
61 0.061 0.834266 0.868179 -2.441797 0.868179 -0.03391
62 0.062 0.831824 0.866182 -2.433472 0.866182 -0.03436
63 0.063 0.82939 0.864191 -2.425171 0.864191 -0.0348
64 0.064 0.826965 0.862205 -2.416896 0.862205 -0.03524
65 0.065 0.824548 0.860223 -2.408645 0.860223 -0.03567
66 0.066 0.82214 0.858247 -2.400419 0.858247 -0.03611
67 0.067 0.819739 0.856275 -2.392218 0.856275 -0.03654
68 0.068 0.817347 0.854308 -2.384041 0.854308 -0.03696
69 0.069 0.814963 0.852346 -2.375889 0.852346 -0.03738
70 0.07 0.812587 0.850389 -2.367761 0.850389 -0.0378
71 0.071 0.810219 0.848437 -2.359658 0.848437 -0.03822
72 0.072 0.80786 0.84649 -2.351579 0.84649 -0.03863
73 0.073 0.805508 0.844548 -2.343524 0.844548 -0.03904
74 0.074 0.803165 0.84261 -2.335494 0.84261 -0.03945
75 0.075 0.800829 0.840677 -2.327487 0.840677 -0.03985
76 0.076 0.798502 0.83875 -2.319505 0.83875 -0.04025
77 0.077 0.796182 0.836826 -2.311546 0.836826 -0.04064
78 0.078 0.793871 0.834908 -2.303612 0.834908 -0.04104
79 0.079 0.791567 0.832994 -2.295701 0.832994 -0.04143
80 0.08 0.789271 0.831085 -2.287814 0.831085 -0.04181
81 0.081 0.786983 0.829181 -2.27995 0.829181 -0.0422
82 0.082 0.784704 0.827281 -2.272111 0.827281 -0.04258
83 0.083 0.782431 0.825387 -2.264294 0.825387 -0.04296
84 0.084 0.780167 0.823496 -2.256501 0.823496 -0.04333
85 0.085 0.777911 0.821611 -2.248732 0.821611 -0.0437
86 0.086 0.775662 0.81973 -2.240986 0.81973 -0.04407
87 0.087 0.773421 0.817854 -2.233263 0.817854 -0.04443
88 0.088 0.771188 0.815982 -2.225563 0.815982 -0.04479
89 0.089 0.768962 0.814115 -2.217886 0.814115 -0.04515
90 0.09 0.766744 0.812253 -2.210233 0.812253 -0.04551
91 0.091 0.764534 0.810395 -2.202602 0.810395 -0.04586
92 0.092 0.762331 0.808542 -2.194994 0.808542 -0.04621
93 0.093 0.760136 0.806693 -2.187409 0.806693 -0.04656
94 0.094 0.757949 0.804849 -2.179847 0.804849 -0.0469
95 0.095 0.755769 0.80301 -2.172307 0.80301 -0.04724
96 0.096 0.753597 0.801174 -2.16479 0.801174 -0.04758
97 0.097 0.751432 0.799344 -2.157296 0.799344 -0.04791
98 0.098 0.749275 0.797518 -2.149824 0.797518 -0.04824
99 0.099 0.747125 0.795696 -2.142375 0.795696 -0.04857
100 0.1 0.744983 0.793879 -2.134948 0.793879 -0.0489
101 0.101 0.742848 0.792066 -2.127543 0.792066 -0.04922
102 0.102 0.74072 0.790258 -2.12016 0.790258 -0.04954
103 0.103 0.7386 0.788454 -2.1128 0.788454 -0.04985
104 0.104 0.736487 0.786654 -2.105461 0.786654 -0.05017
105 0.105 0.734382 0.784859 -2.098145 0.784859 -0.05048
106 0.106 0.732283 0.783069 -2.09085 0.783069 -0.05079
107 0.107 0.730193 0.781282 -2.083578 0.781282 -0.05109
108 0.108 0.728109 0.7795 -2.076327 0.7795 -0.05139
109 0.109 0.726033 0.777722 -2.069098 0.777722 -0.05169
110 0.11 0.723964 0.775949 -2.061891 0.775949 -0.05199
111 0.111 0.721902 0.77418 -2.054705 0.77418 -0.05228
112 0.112 0.719847 0.772415 -2.047541 0.772415 -0.05257
113 0.113 0.717799 0.770655 -2.040398 0.770655 -0.05286
114 0.114 0.715759 0.768899 -2.033277 0.768899 -0.05314
115 0.115 0.713726 0.767147 -2.026177 0.767147 -0.05342
116 0.116 0.7117 0.765399 -2.019099 0.765399 -0.0537
117 0.117 0.709681 0.763656 -2.012042 0.763656 -0.05398
118 0.118 0.707668 0.761917 -2.005005 0.761917 -0.05425
119 0.119 0.705663 0.760182 -1.99799 0.760182 -0.05452
120 0.12 0.703665 0.758451 -1.990996 0.758451 -0.05479
121 0.121 0.701674 0.756724 -1.984023 0.756724 -0.05505
122 0.122 0.69969 0.755002 -1.977071 0.755002 -0.05531
123 0.123 0.697713 0.753284 -1.97014 0.753284 -0.05557
124 0.124 0.695743 0.751569 -1.96323 0.751569 -0.05583
125 0.125 0.69378 0.74986 -1.95634 0.74986 -0.05608
126 0.126 0.691824 0.748154 -1.949471 0.748154 -0.05633
127 0.127 0.689874 0.746452 -1.942623 0.746452 -0.05658
128 0.128 0.687932 0.744754 -1.935795 0.744754 -0.05682
129 0.129 0.685996 0.743061 -1.928987 0.743061 -0.05706
130 0.13 0.684067 0.741371 -1.9222 0.741371 -0.0573
131 0.131 0.682145 0.739686 -1.915434 0.739686 -0.05754
132 0.132 0.680229 0.738004 -1.908687 0.738004 -0.05778
133 0.133 0.67832 0.736327 -1.901961 0.736327 -0.05801
134 0.134 0.676419 0.734654 -1.895256 0.734654 -0.05824
135 0.135 0.674523 0.732985 -1.88857 0.732985 -0.05846
136 0.136 0.672635 0.731319 -1.881904 0.731319 -0.05868
137 0.137 0.670753 0.729658 -1.875258 0.729658 -0.05891
138 0.138 0.668878 0.728001 -1.868633 0.728001 -0.05912
139 0.139 0.667009 0.726347 -1.862027 0.726347 -0.05934
140 0.14 0.665147 0.724698 -1.855441 0.724698 -0.05955
141 0.141 0.663291 0.723052 -1.848874 0.723052 -0.05976
142 0.142 0.661443 0.721411 -1.842328 0.721411 -0.05997
143 0.143 0.6596 0.719773 -1.835801 0.719773 -0.06017
144 0.144 0.657764 0.71814 -1.829293 0.71814 -0.06038
145 0.145 0.655935 0.71651 -1.822805 0.71651 -0.06057
146 0.146 0.654112 0.714884 -1.816337 0.714884 -0.06077
147 0.147 0.652296 0.713262 -1.809888 0.713262 -0.06097
148 0.148 0.650486 0.711644 -1.803458 0.711644 -0.06116
149 0.149 0.648683 0.710029 -1.797048 0.710029 -0.06135
150 0.15 0.646886 0.708419 -1.790657 0.708419 -0.06153
151 0.151 0.645095 0.706812 -1.784285 0.706812 -0.06172