Table 1 | |||||
Initial x | Initial y | Stepsize h | Appx. y | Exa. y | Error |
0.00000 | 1.00000 | 0.10000 | |||
0.10000 | 1.00000 | 0.10000 | 1.10000 | 1.11111 | 0.01111 |
0.20000 | 1.10000 | 0.10000 | 1.22100 | 1.25000 | 0.02900 |
0.30000 | 1.22100 | 0.10000 | 1.37008 | 1.42857 | 0.05849 |
0.40000 | 1.37008 | 0.10000 | 1.55780 | 1.66667 | 0.10887 |
0.50000 | 1.55780 | 0.10000 | 1.80047 | 2.00000 | 0.19953 |
Table 2 | |||||
Initial x | Initial y | Stepsize h | Appx. y | Exa. y | Error |
0.00000 | 1.00000 | 0.05000 | |||
0.05000 | 1.00000 | 0.05000 | 1.05000 | 1.05263 | 0.00263 |
0.10000 | 1.05000 | 0.05000 | 1.10513 | 1.11111 | 0.00599 |
0.15000 | 1.10513 | 0.05000 | 1.16619 | 1.17647 | 0.01028 |
0.20000 | 1.16619 | 0.05000 | 1.23419 | 1.25000 | 0.01581 |
0.25000 | 1.23419 | 0.05000 | 1.31035 | 1.33333 | 0.02298 |
0.30000 | 1.31035 | 0.05000 | 1.39620 | 1.42857 | 0.03237 |
0.35000 | 1.39620 | 0.05000 | 1.49367 | 1.53846 | 0.04479 |
0.40000 | 1.49367 | 0.05000 | 1.60522 | 1.66667 | 0.06144 |
0.45000 | 1.60522 | 0.05000 | 1.73406 | 1.81818 | 0.08412 |
0.50000 | 1.73406 | 0.05000 | 1.88441 | 2.00000 | 0.11559 |
Table 3 | |||||
Initial x | Initial y | Stepsize h | Appx. y | Exa. y | Error |
0.00000 | 1.00000 | 0.02500 | |||
0.02500 | 1.00000 | 0.02500 | 1.02500 | 1.02564 | 0.00064 |
0.05000 | 1.02500 | 0.02500 | 1.05127 | 1.05263 | 0.00137 |
0.07500 | 1.05127 | 0.02500 | 1.07889 | 1.08108 | 0.00219 |
0.10000 | 1.07889 | 0.02500 | 1.10799 | 1.11111 | 0.00312 |
0.12500 | 1.10799 | 0.02500 | 1.13869 | 1.14286 | 0.00417 |
0.15000 | 1.13869 | 0.02500 | 1.17110 | 1.17647 | 0.00537 |
0.17500 | 1.17110 | 0.02500 | 1.20539 | 1.21212 | 0.00673 |
0.20000 | 1.20539 | 0.02500 | 1.24171 | 1.25000 | 0.00829 |
0.22500 | 1.24171 | 0.02500 | 1.28026 | 1.29032 | 0.01006 |
0.25000 | 1.28026 | 0.02500 | 1.32124 | 1.33333 | 0.01210 |
0.27500 | 1.32124 | 0.02500 | 1.36488 | 1.37931 | 0.01443 |
0.30000 | 1.36488 | 0.02500 | 1.41145 | 1.42857 | 0.01712 |
0.32500 | 1.41145 | 0.02500 | 1.46125 | 1.48148 | 0.02023 |
0.35000 | 1.46125 | 0.02500 | 1.51464 | 1.53846 | 0.02383 |
0.37500 | 1.51464 | 0.02500 | 1.57199 | 1.60000 | 0.02801 |
0.40000 | 1.57199 | 0.02500 | 1.63377 | 1.66667 | 0.03290 |
0.42500 | 1.63377 | 0.02500 | 1.70050 | 1.73913 | 0.03863 |
0.45000 | 1.70050 | 0.02500 | 1.77279 | 1.81818 | 0.04539 |
0.47500 | 1.77279 | 0.02500 | 1.85136 | 1.90476 | 0.05340 |
0.50000 | 1.85136 | 0.02500 | 1.93705 | 2.00000 | 0.06295 |
Table 4 | |||||
Initial x | Initial y | Stepsize h | Appx. y | Exa. y | Error |
0 | 1 | 0.0250000 | |||
0.0250000 | 1 | 0.0250000 | 1.025 | 1.025641026 | 0.000641026 |
0.0500000 | 1.025 | 0.0250000 | 1.051265625 | 1.052631579 | 0.001365954 |
0.0750000 | 1.051265625 | 0.0250000 | 1.07889461 | 1.081081081 | 0.002186471 |
0.1000000 | 1.07889461 | 0.0250000 | 1.10799495 | 1.111111111 | 0.003116161 |
0.1250000 | 1.10799495 | 0.0250000 | 1.13868627 | 1.142857143 | 0.004170873 |
0.1500000 | 1.13868627 | 0.0250000 | 1.171101431 | 1.176470588 | 0.005369158 |
0.1750000 | 1.171101431 | 0.0250000 | 1.205388395 | 1.212121212 | 0.006732817 |
0.2000000 | 1.205388395 | 0.0250000 | 1.241712424 | 1.25 | 0.008287576 |
0.2250000 | 1.241712424 | 0.0250000 | 1.280258668 | 1.290322581 | 0.010063913 |
0.2500000 | 1.280258668 | 0.0250000 | 1.321235224 | 1.333333333 | 0.012098109 |
0.2750000 | 1.321235224 | 0.0250000 | 1.364876787 | 1.379310345 | 0.014433558 |
0.3000000 | 1.364876787 | 0.0250000 | 1.411449003 | 1.428571429 | 0.017122425 |
0.3250000 | 1.411449003 | 0.0250000 | 1.46125371 | 1.481481481 | 0.020227771 |
0.3500000 | 1.46125371 | 0.0250000 | 1.514635271 | 1.538461538 | 0.023826268 |
0.3750000 | 1.514635271 | 0.0250000 | 1.571988271 | 1.6 | 0.028011729 |
0.4000000 | 1.571988271 | 0.0250000 | 1.633766949 | 1.666666667 | 0.032899718 |
0.4250000 | 1.633766949 | 0.0250000 | 1.70049681 | 1.739130435 | 0.038633625 |
0.4500000 | 1.70049681 | 0.0250000 | 1.772789045 | 1.818181818 | 0.045392773 |
0.4750000 | 1.772789045 | 0.0250000 | 1.85135857 | 1.904761905 | 0.053403335 |
0.5000000 | 1.85135857 | 0.0250000 | 1.937046784 | 2 | 0.062953216 |
We conclude this section with a flow chart for Euler's Algorithm.