Question

Solve the following first order ODE with a given initial condition using Euler method in Excel using the formula given with n= 3, 10, and 100: y(n1)y(n)f(x(n), y(n)). dx (b-a) dx y'(x(n), y (n)) y'6where y (3) = 1 on the interval [3,6] b.y'yinwhere y (2)= e on the interval [2,5] a. Create a table for each n-values given and a graph one separately.

Answer 1

Using Excel and given differential equation and by Euler a) the calculations are tabulated as follow. Case-1: n 3, then dx-1. a-3,b-6 f(x(n),y(n)) у(п) x(п) dx x(n+1) yn+1) n+1 C 3 1 1 5.333333333 1 4 6.333333 4 2.833333333 1 6.333333 5 9.166667 1 5 9.166667 1 2.333333333 3 11.5 6 1 11.5 2.166666667 13.66667 Its summary Table is x(п) y(n) 3 1 4 6.333333 9.166667 6 11.5 Its graph is y(n) 14 12 10 v(n) 4 0 6

Case-2: n 10, then dx 0.3 a-3,b-6 y(n+1) x(п) y(n) f(x(n),y(n)) x(n+1) dx n+1 r C 3 1 0.3 5.333333333 1 3.3 2.6 3.3 3.927273 2,6 0.3 4.424242424 3.6 0 3 2 3.6 3.927273 3.818181818 3 9 5.072727 5.072727 2 3.9 0.3 3.398601399 4 4.2 6.092308 4 4.2 6.092308 0.3 3.098901099 4.5 7.021978 01 4.5 7,021978 2.879120879 6 1.8 7.885714 4.8 7.885714 0.3 2.714285714 5.1 8.7 2.588235294 5,1 0 3 5.4 8.7 9.476471 5,4 9 9.476471 0 2.490196078 5.7 10.22353 2.412796698 5.7 10.22353 0.3 10 6 10.94737 Its summary table is x(п) y(n) 3 3.3 2.6 3.927273 ,6 .9 5.072727 1.2 6.092308 4.5 7.021978 4 1 7.885714 5.1 8.7 5 4 9.476471 57 10.22353 10.94737 Its graph is y(n) 12 10 yin) 1 2 4

Case-3: n 100, then dx-0.03 a-3,b- 6 y(n) f(x(n),y(n)) x(n) dx n+1 x(n+1) y(n+1) n 5.333333333 0 3 1 0.03 1 3.03 1.16 3.03 5.234323432 1 1.16 0.03 2 3.06 1.31703 3.06 1.31703 0.03 5.139196273 3 3.09 1.471206 0.03 3 3.09 1.471206 5.047763371 4 3.12 1.622638 3,12 4 1.622638 0.03 4.95984712 5 3.15 1.771434 5 3.15 1.771434 0.03 4.87528006 6 3.18 1.917692 4.793904209 6 3.18 1.917692 0.03 7 3.21 2.061509 4.715570446 7 3.21 2.061509 0.03 3.24 2.202977 8 3.24 2.342181 2.202977 0.03 4.640137934 3.27 3.27 2.342181 0.03 4.567473587 10 3.3 2.479205 10 2.479205 4.49745158 3.3 0.03 3.33 2.614128 3.33 2.614128 0.03 12 11 4.429952889 3.36 2.747027 12 3.36 2.747027 0.03 4.364864865 13 3.39 2.877973 13 14 3.39 2.877973 0.03 4.302080842 3.42 3.007035 0.03 15 14 3.42 3.007035 4.241499767 3.45 3.13428 15 3.45 3.13428 0.03 4.18302586 16 3,48 3.259771 4.126568295 16 3.48 3.259771 0.03 17 3.51 3.383568 4.072040903 17 3.51 3.383568 0.03 18 3.54 3.505729 18 3.54 3.505729 0.03 4.019361897 19 3.57 3.62631 19 3.57 3.62631 0.03 3.968453614 20 3.6 3.745364 20 3.6 3.745364 0.03 3.919242273 21 3.63 3.862941 21 3,63 0.03 3.871657754 3.66 3.979091 3.862941 22 3.825633383 22 3.66 3.979091 0.03 23 3.69 4.09386 23 3.69 4.09386 0.03 3.78110574 24 3.72 4,207293 24 3.72 0.03 3.738014472 25 4.207293 3.75 4.319434 3.696302124 25 3.75 4.319434 0.03 26 3.78 4.430323 3.78 4.430323 0.03 27 3.81 26 3.655913978 4.54 27 3.81 4,54 0.03 28 3.6167979 3.84 4.648504 28 3.84 4.648504 0.03 3.578904199 29 3.87 4.755871 29 3.87 4.755871 0.03 3.542185497 30 3.9 4.862137 30 3.9 0.03 3.506596601 4.862137 31 3.93 4.967335 3.472094389 31 3.93 4.967335 0.03 32 3.96 5.071497 3.96 33 32 5.071497 0.03 3.438637698 3.99 5.174656 33 3.99 3.406187224 34 4.02 5.276842 5.174656 0.03 3.37470542 34 4.02 5.276842 0.03 35 4.05 5.378083 36 35 4.05 5.378083 0.03 3.344156411 4.08 5.478408 4.08 3.314505902 36 5.478408 0.03 37 4.11 5.577843 37 3.285721101 4,11 5.577843 0.03 38 4.14 5.676415 38 4.14 0.03 39 4.17 5.676415 3.257770642 5.774148 39 4.17 5.774148 0.03 3.230624513 40 4.2 5.871067 4C 4.2 5.871067 0.03 41 4.23 3.204253988 5.967194

4 3.178631563 41 4,23 5.967194 0.03 42 4.26 6.062553 42 4,26 6.062553 0.03 3.153730896 43 4.29 6.157165 43 4.29 3.129526751 4.32 6.157165 0.03 44 6.251051 44 4,32 6.251051 0.03 .105994944 45 4.35 6.344231 45 4,35 4.38 6.344231 0.03 3.08311229 46 6.436724 46 4.38 6.436724 0.03 3.060856558 47 4.41 6.52855 47 4,41 0.03 4.44 6.52855 3.039206424 48 6.619726 48 4.44 6.619726 0,03 3.018141429 49 4.47 6.71027 49 4.47 6.71027 0.03 2.997641937 50 4.5 6.8002 2.977689098 50 4.5 6.8002 0.03 51 4.53 6.88953 51 4,53 0.03 2.958264812 52 4.56 6.88953 6.978278 4,56 2,93935169 4,59 52 6.978278 0.03 53 7.066459 2.92093303 53 4.59 7.066459 0.03 54 4.62 7.154087 54 4.62 2.902992776 4.65 7.154087 0.03 55 7.241176 55 7.241176 56 4,68 4.65 0.03 2.885515497 7,327742 56 4.68 7.327742 0,03 2.868486352 57 4.71 7.413797 57 4.71 7,413797 0.03 2.851891072 58 4.74 7,499353 4.77 58 4,74 7.499353 0.03 2.835715925 59 7.584425 59 4.77 7.584425 0.03 2.8199477 60 4.8 7.669023 2.80457368 60 4.8 7.669023 0.03 61 4.83 7.75316 61 4.83 7,75316 0.03 2.789581624 62 4.86 7,836848 62. 4.86 7.836848 0.03 2.774959742 63 4,89 7.920097 63 4.89 2.76069668 7.920097 0.03 64 4,92 8.002918 64 4.92 8.002918 0.03 2.746781496 65 4.95 8.085321 65 4.95 66 8.085321 0.03 2.733203651 4,98 8.167317 66 5.01 8.248916 4,98 8.167317 0,03 2.719952983 67 67 5.01 8.248916 0.03 2.707019696 68 5.04 8.330126 68 5.04 0.03 2.694394344 69 5.07 8.410958 8.330126 69 5.07 8.410958 2.682067817 70 0.03 5.1 8.49142 70 5.13 5.1 8.49142 0.03 2.670031326 71 8.571521 71 5.13 8.571521 0.03 2.658276391 72 5.16 8.651269 72 5.16 2.646794826 5.19 8.651269 0.03 73 8.730673 5,19 2,63557873 74 5.22 73 8.730673 0.03 8.809741 74 5.25 5.22 8.809741 0,03 2.624620476 75 8.888479 75 5.25 8.888479 0.03 2.613912697 76 5.28 8.966897 76 5.28 8.966897 0.03 2.603448276 77 5.31 9.045 77 2.593220339 5.31 9.045 0.03 78 5.34 9.122797 78 5.34 9.122797 0.03 2.583222243 79 5.37 9.200293 79 5.37 9.200293 0.03 80 5.4 2.573447569 9.277497 80 0,03 2.563890109 5.43 5.4 9.277497 81 9.354413 81 5.43 9.354413 0.03 2.554543865 82 5.46 9.43105 82 5.46 9.43105 0.03 2.545403032 83 5.49 9.507412 83 5,49 9.507412 0.03 84 5.52 2.536461998 9.583506 85 5.55 9,659337 84 5.52 9.583506 0.03 2.527715335 csSCTAC 86 85 5.55 5.58 9.659337 0.03 2.519157789 9.734912 86 5.58 9.734912 0.03 2.510784277 87 5.61 9.810235

2.502589876 87 5.61 9.810235 0.03 88 5.64 9.885313 9.96015 88 5.64 9.885313 0.03 2.494569825 89 5.67 89 2.48671951 5.67 9.96015 0.03 90 5.7 10.03475 90 5.7 10.03475 0.03 2.479034466 91 5.73 10.10912 91 5.73 10.10912 0.03 2.471510364 92 5.76 10.18327 92 5.76 10.18327 0.03 2.464143015 93 5.79 10.25719 93 5.79 10.25719 0.0 2.456928356 94 5.82 10.3309 94 5.82 10.3309 0.03 2.449862454 95 5.85 10.4044 95 5.85 10.4044 0.03 2.442941493 96 5.88 10.47768 96 5.88 10.47768 0.03 2.436161776 97 5.91 10.55077 97 5.92 10.55077 0.03 2.429519719 98 5.94 10.62365 98 10.69635 5.94 10.62365 0.03 2.423011844 99 5.97 2.416634781 5.97 10.69635 0.03 100 6 10.76884 Its summary table is x(п) y(n) x(n) у(n) х(п) y(n) x(n) y(n) x(п) у(n) 3 1 3.6 3.745364 4.2 5.871067 4.8 7.669023 54 9.277497 3.03 1.16 3.63 3.862941 4.23 5.967194 4.83 7.75316 5.43 9.354413 3.06 1.31703 3.66 3.979091 4.26 6.062553 4.86 7.836848 5.46 9.43105 3.09 1.471206 3.69 4,09386 4.29 6.157165 4.89 7.920097 5.49 9.507412 3.12 1.622638 3.72 4.207293 4.32 6.251051 4.92 8.002918 5.52 9.583506 6.344231 3.15 1.771434 3.75 4.319434 4.35 4.95 8.085321 5.55 9.659337 3.18 1.917692 3.78 4.430323 4.38 6.436724 4.98 8.167317 5.58 9.734912 3.21 2.061509 3.81 4.54 4.41 6.52855 5.01 8.248916 5.61 9.810235 3.24 2.202977 3.84 4.648504 4.44 6.619726 5.04 8.330126 5.64 9.885313 3.27 2.342181 3.87 4.755871 4.47 6.71027 5.07 8.410958 5.67 9.96015 6.8002 10.03475 3.3 2.479205 3.9 4.862137 4.5 5.1 8.49142 5.7 3.33 2.614128 3.93 4.967335 4.53 6.88953 5.13 8.571521 5.73 10.10912 3.36 2.747027 3.96 5.071497 4,56 6.978278 5.16 8.651269 5.76 10.18327 3.39 2.877973 3.99 5.174656 4.59 7.066459 5.19 8.730673 5.79 10.25719 3.42 3.007035 4.02 5.276842 4.62 7.154087 5.22 8.809741 5.82 10.3309 5.378083 7.241176 10.4044 3.45 3.13428 4.05 4.65 5.25 8.888479 5.85 5.478408 3.48 3.259771 4.08 4.68 7.327742 5.28 8.966897 5.88 10.47768 3.51 3.383568 4.11 5.577843 4.71 7.413797 5.31 9.045 5.91 10.55077 3.54 3.505729 4.14 5.676415 4,74 7.499353 5.34 9.122797 5.94 10.62365 5.774148 3.57 3.62631 4.17 4,77 7.584425 5.37 9.200293 5.97 10.69635 6 10.76884 Its graph is

6 У(n) 12 10 y(n) 4 1 2 3 Using Excel and given differential equation and by Euler b) the calculations are tabulated as follow. Case-1: n-3, dx-1, a- 2,b -5 x(n) y(n) f(x(n),y(n)) dx n+1 x(n+1) y(n+1) 3 2 2.718 1 1.358859 4,076859 4 1 3 4.076859 1 1.909773 5.986632 5.986632 2.678313 5 2 4 1 3 8.664945 8.664945 12.40696 5 1 3.742018 4 6 Its summary table is x(n) у(n) 2 2.718 3 4.076859 4 5.986632 5 8.664945 Its graph is

y(n) 10 y(n) 2 1 0 2 3 5 6 1 Case-2: n 10 a-2,b-5 f(x(n),y(n)) x(п) y(n) dx x(n+1) y(n+1) n+1 0 0.3 1.358859 1 125658 2.3 3.125658 0.3 1.548756 2 2.6 3.590285 2.6 3.590285 0.3 1.765083 2.9 4.119809 2.9 0.3 3.2 4.119809 2.011329 4 4.723208 3.5 3.2 4.723208 0.3 2.291477 5.410651 5.410651 5 3.5 03 2.610051 3.8 6.193666 3.8 6.193666 0.3 2.972189 7 4.1 7.085323 0.3 4.4 4.1 7.085323 3.383718 8.100439 4.7 4.4 8.100439 0.3 3.85124 9 9.25581 4.382235 10.57048 л- 9.25581 0.3 10 5 Its summary table is x(п) y(n) 2.718 2.3 3.125658 2.6 3.590285 2.9 4.119809 3.2 4.723208 3.5 5.410651 3.8 6.193666 4.1 7.085323 4.4 8.100439 4.7 9.25581 5 10.57048 Its graph is

y(n) 12 10 8 y(n) A 2 0 6 1 4 Case-3: n 100, dx-0.03, a- 2, b=5 y(n) x(n+1) x(n) dx f(x(n),y(n) n+1 y(n+1) n 0 2,03 2.718 0.03 1.358859 1 2.758766 2.758766 2.800138 1 2.03 0.03 1.379089 2 2.06 2,800138 2 2.06 0.03 1.399619 2.09 2.842127 3 2.09 2.842127 0.03 1.420455 4 2.12 2.884741 0.03 |441602 2.15 4 2.12 2.884741 2.927989 2.15 0.03 2.18 2.927989 1.463062 6 2.971881 2.971881 3.016426 2.18 0.03 1.484843 7 2,21 7 2.21 3.016426 0.03 1.506947 8 2.24 3.061634 2.24 3.061634 0.03 1.52938 9 2.27 3.107516 9 2.3 2.27 3.107516 0.03 1.552 147 10 3.15408 10 0.03 1.575253 2.33 3.201338 2.3 3.15408 11 3.201338 11 2.33 0.03 1.598702 12 2.36 3.249299 12 0.03 1.6225 2.39 2.36 3.249299 13 3.297974 2.42 13 2.39 3.297974 0.03 .646652 14 3.347373 14 0,03 2,45 2,42 3.347373 1.671164 15 3.397508 3,397508 15 2,45 0.03 .696039 16 2.48 3.448389 16 2.48 0.03 3.448389 1.721285 17 2.51 3.500028 17 2.51 0.03 1,746906 2.54 3.500028 18 3.552435 18 2.54 3.552435 0.03 1.772908 19 2.57 3.605622 ,605622 19 0.03 1.799296 2.57 20 2.6 3.659601 20 2.6 3.659601 0.03 1.826077 21 2.63 3.714384 2.66 21 2.63 3.714384 0.03 1.853255 22 3.769981 22 2.66 0.03 23 2.69 3.769981 1.880838 3.826406 23 3.826406 2.69 0.03 1.908831 24 2.72 3.883671 24 2,75 2.72 3.883671 0.03 1.937239 25 3.941788 25 2.75 3.941788 0.03 1.96607 26 2.78 4.000771 26 2.78 0,03 1.99533 27 2.81 4.000771 4,06063 t

9 27 2.81 4.06063 0.03 2,025024 28 2.84 4.121381 28 2,84 0.03 29 2.87 4.121381 2.055159 4.183036 29 2.87 4.183036 0.03 2.085742 30 2.9 4.245608 30 2.9 4.245608 0.03 2.11678 31 2.93 4.309112 31 2.93 4.309112 0.03 2.148279 32 2.96 4.37356 32 2.180245 33 2.96 4.37356 0.03 2.99 4.438967 33 2.99 4.438967 0.03 2.212687 34 3.02 4.505348 34 3.02 4.505348 0.03 2.24561 35 3.05 4.572716 35 3.05 36 3.08 4.572716 0.03 2.279023 4.641087 36 4.641087 2.312932 37 3.08 0.03 3.11 4.710475 4.710475 37 3.11 0.03 2.347344 38 3.14 4.780895 38 3.14 4.780895 0.03 2.382268 39 3.17 4.852363 39 3.17 40 3.2 4.852363 0.03 2.41771 4.924895 40 3.2 4.924895 0.03 2,453679 41 3.23 4.998505 41 3.23 4.998505 0.03 2.490182 42 3.26 5.07321 42 3.26 5.07321 0.03 2.527227 43 3.29 5.149027 43 3.29 44 5.149027 0.03 2.564823 3.32 5.225972 44 5.225972 3.32 0.03 2.602976 45 3.35 5.304061 45 46 5.383312 3.35 5.304061 0.03 2.641696 3.38 46 3.38 0.03 2.680991 47 5.383312 3.41 5.463742 47 5.545368 3.41 5.463742 0.03 2.72087 48 3.44 48 49 3.44 5.545368 0.03 2.76134 3.47 5.628208 49 3.47 5.628208 0.03 2.802412 50 3.5 5.71228 50 3.5 5,71228 51 5.797603 0.03 2.844093 3.53 51 5.797603 3.53 0.03 2.886393 52 3.56 5.884195 3.56 52 5.884195 0.03 2.929321 53 3.59 5.972075 53 3.59 5.972075 0.03 2.972886 54 3.62 6.061261 54 3.62 6.061261 0.03 3.017098 55 3.65 6.151774 55 3.65 6.151774 0.03 3.061966 56 3.68 6.243633 56 3.68 6.243633 0.03 3.107501 57 3.71 6.336858 57 3.71 0.03 58 3.74 6.336858 3.153711 6.431469 58 59 6.527488 3.74 6.431469 0.03 3.200607 3.77 59 3,77 6.527488 0.03 3.248199 60 3.8 6.624934 60 3.8 6.624934 0.03 3.296498 61 3.83 6.723829 61 62 3.86 3.83 6.723829 0.03 3.345513 6.824194 62 3.86 6.824194 0.03 3.395256 63 3.89 6.926052 63 64 3.89 6.926052 0.03 3.445737 3.92 7.029424 64 3.92 7.029424 0.03 3.496968 65 3.95 7.134333 65 3.548958 66 3.95 7.134333 0.03 3.98 7.240802 66 67 3.98 7.240802 0.03 3.60172 4.01 7.348853 68 67 4.01 7.348853 0.03 3.655265 4.04 7.458511 68 4.04 69 7.569799 7.458511 0.03 3.709605 4.07 69 4.07 70 7.569799 0.03 3.764751 4.1 7.682742 70 71 4.1 7.682742 0.03 3.820715 4.13 7.797363 71 72 4.16 4.13 7.797363 0.03 3.877509 7.913689 72 3.935146 8.031743 4.16 7.913689 0.03 73 4.19

10 73 8,031743 4.22 4.19 0,03 3.993639 74 8.151552 74 4.25 4.22 8.151552 0.03 4.052999 75 8.273142 4.28 75 4.25 8.273142 0.03 4.11324 76 8.396539 76 4.28 77 4.31 8.396539 0.03 4.174374 8.52177 77 4,34 8,648863 4.31 8.52177 0.03 4,236416 78 8.648863 78 4.34 0.03 4.299377 79 4.37 8.777844 4.37 79 8.777844 0.03 4.363273 80 4.4 8.908742 80 4.4 4.428117 4.43 8.908742 0.03 81 9.041586 81 4,43 0,03 82 4,46 9.041586 4.493923 9.176404 82 4.46 9.176404 0.03 4.560704 83 4.49 9.313225 83 4.52 4.49 9.313225 0.03 4.628476 84 9.452079 84 4.52 0.03 4.697254 85 4,55 9.452079 9.592997 85 4.55 9.592997 0.03 4.767052 86 4.58 9.736008 4.837885 9.881145 86 4.58 9.736008 0.03 87 4.61 87 4.61 9.881145 0.03 4.909768 88 4.64 10.02844 88 4.64 10.02844 0.03 4.982718 89 4.67 10.17792 4.67 89 10.17792 0.03 5.056749 90 4.7 10.32962 10.48358 90 4.7 4.73 10.32962 0.03 5.131878 91 91 92 4,76 4.73 10.48358 0.03 5.208122 10.63982 4.79 92 4.76 10.63982 0.03 5.285496 93 10.79839 93 0.03 4.79 10.79839 5.364018 94 4.82 10.95931 94 4,82 10.95931 0,03 5.443704 95 4,85 11.12262 95 4.85 11.12262 0.03 5.524572 96 4.88 11.28836 96 4.88 97 11.28836 0.03 5.606639 4.91 11.45655 97 4,91 0.03 11.45655 5.689922 98 4,94 11.62725 98 4,94 11.62725 0.03 5.774441 99 4,97 11.80049 99 4.97 11.80049 0.03 5.860213 100 5 11.97629 100 11.97629 0.03 5.947257 5.03 5 101 12.15471 Its summary table: x(п) y(n) x(п) у(n) x(п) x(п) y(n) y(n) x(п) у(n) 2 2.718 2.6 3.659601 3.2 4.924895 3.8 6.624934 4.4 8.908742 2.03 2.63 3.23 3.83 4,43 9,041586 2.758766 3.714384 4.998505 6.723829 6.824194 2.06 2.800138 2.66 3.769981 3.26 5.07321 3.86 4.46 9.176404 2.09 2.842127 2.69 3.826406 3.29 5.149027 3.89 6.926052 4.49 9.313225 2.12 2.72 3.32 4.52 2.884741 3.883671 5.225972 3.92 7.029424 9.452079 2.15 2.927989 2.75 3.941788 3.35 5.304061 3.95 7.134333 4,55 9.592997 2.18 2.971881 2.78 4.000771 3.38 5.383312 3.98 7.240802 4.58 9.736008 2.21 2.81 3.41 3.016426 4.06063 5.463742 4.01 7.348853 4.61 9.881145 2.24 3.061634 2.84 3.44 4.04 7.458511 4.64 10.02844 4,121381 5.545368 2.27 3.107516 2.87 4.183036 3.47 5.628208 4.07 7.569799 4.67 10.17792 2.3 3.15408 2.9 4.245608 3.5 5.71228 4.1 7.682742 4.7 10.32962 2.33 2.93 3.53 3.201338 4.309112 5.797603 4.13 7.797363 4.73 10.48358 7.913689 2.36 2.96 3.249299 4,37356 3.56 5.884195 4.16 4.76 10.63982

11 5.972075 2.39 3.297974 2.99 4.438967 3.59 4.19 8.031743 4.79 10.79839 6.061261 10.95931 2,42 3.347373 3.02 4.505348 3.62 4.22 8.151552 4.82 2.45 3.397508 3.05 4.572716 3.65 6.151774 4.25 8.273142 4.85 11.12262 2.48 3.08 3.68 4.28 4.88 11.28836 3.448389 4.641087 6.243633 8.396539 2.51 3.500028 3.11 4.710475 3.71 6.336858 4.31 8.52177 4.91 11.45655 2.54 11.62725 3.552435 3.14 4.780895 3.74 6.431469 4.34 8.648863 4.94 2.57 3.605622 3.17 4.852363 3.77 6.527488 4.37 8.777844 4.97 11.80049 11.97629 5 Its graph is y(n) 14 12 10 8 vin) 2 0