Question

2. Now let's investigate how the various methods work when applied to an especially simple differential equation, x' x (a) First find the explicit solution x(t) of this equation satisfying the initial condition x(0) = 1 (now there's a free gift from the math department... (b) Now use Euler's method to approximate the value of x(1)e using the step size At = 0.1. That is, recursively determine tk and xk for k 1,.., 10 using At = 0.1 and starting with to = 0 and xo (c) Repeat the previous step with At half the size, namely, 0.05 (d) Again use Euler's method, this time reducing the step size by a factor of 5, so that At = 0.01, to approximate x(1) - = 1.

Answer 1

J을- Jot oly Jnlx) tC is here Cis conslant U stng iniHral Cand Euler Melhoc. PCtu plkt ncxF F 2 A.21 O 3 O.u 0.5 1.141G 1 1-g487 )2. S 93 0. 2-43 d. 2.3579 2-5937

(c)

k tk xk
1 0.050000 1.050000   
2 0.100000 1.102500   
3 0.150000 1.157625   
4 0.200000 1.215506   
5 0.250000 1.276282   
6 0.300000 1.340096   
7 0.350000 1.407100   
8 0.400000 1.477455   
9 0.450000 1.551328   
10 0.500000 1.628895   
11 0.550000 1.710339   
12 0.600000 1.795856   
13 0.650000 1.885649   
14 0.700000 1.979932   
15 0.750000 2.078928   
16 0.800000 2.182875   
17 0.850000 2.292018   
18 0.900000 2.406619   
19 0.950000 2.526950   
20 1.000000 2.653298   

x(1)=2.653298

(d)

k tk xk
1 0.010000 1.010000   
2 0.020000 1.020100   
3 0.030000 1.030301   
4 0.040000 1.040604   
5 0.050000 1.051010   
6 0.060000 1.061520   
7 0.070000 1.072135   
8 0.080000 1.082857   
9 0.090000 1.093685   
10 0.100000 1.104622   
11 0.110000 1.115668   
12 0.120000 1.126825   
13 0.130000 1.138093   
14 0.140000 1.149474   
15 0.150000 1.160969   
16 0.160000 1.172579   
17 0.170000 1.184304   
18 0.180000 1.196147   
19 0.190000 1.208109   
20 0.200000 1.220190   
21 0.210000 1.232392   
22 0.220000 1.244716   
23 0.230000 1.257163   
24 0.240000 1.269735   
25 0.250000 1.282432   
26 0.260000 1.295256   
27 0.270000 1.308209   
28 0.280000 1.321291   
29 0.290000 1.334504   
30 0.300000 1.347849   
31 0.310000 1.361327   
32 0.320000 1.374941   
33 0.330000 1.388690   
34 0.340000 1.402577   
35 0.350000 1.416603   
36 0.360000 1.430769   
37 0.370000 1.445076   
38 0.380000 1.459527   
39 0.390000 1.474123   
40 0.400000 1.488864   
41 0.410000 1.503752   
42 0.420000 1.518790   
43 0.430000 1.533978   
44 0.440000 1.549318   
45 0.450000 1.564811   
46 0.460000 1.580459   
47 0.470000 1.596263   
48 0.480000 1.612226   
49 0.490000 1.628348   
50 0.500000 1.644632   
51 0.510000 1.661078   
52 0.520000 1.677689   
53 0.530000 1.694466   
54 0.540000 1.711410   
55 0.550000 1.728525   
56 0.560000 1.745810   
57 0.570000 1.763268   
58 0.580000 1.780901   
59 0.590000 1.798710   
60 0.600000 1.816697   
61 0.610000 1.834864   
62 0.620000 1.853212   
63 0.630000 1.871744   
64 0.640000 1.890462   
65 0.650000 1.909366   
66 0.660000 1.928460   
67 0.670000 1.947745   
68 0.680000 1.967222   
69 0.690000 1.986894   
70 0.700000 2.006763   
71 0.710000 2.026831   
72 0.720000 2.047099   
73 0.730000 2.067570   
74 0.740000 2.088246   
75 0.750000 2.109128   
76 0.760000 2.130220   
77 0.770000 2.151522   
78 0.780000 2.173037   
79 0.790000 2.194768   
80 0.800000 2.216715   
81 0.810000 2.238882   
82 0.820000 2.261271   
83 0.830000 2.283884   
84 0.840000 2.306723   
85 0.850000 2.329790   
86 0.860000 2.353088   
87 0.870000 2.376619   
88 0.880000 2.400385   
89 0.890000 2.424389   
90 0.900000 2.448633   
91 0.910000 2.473119   
92 0.920000 2.497850   
93 0.930000 2.522829   
94 0.940000 2.548057   
95 0.950000 2.573538   
96 0.960000 2.599273   
97 0.970000 2.625266   
98 0.980000 2.651518   
99 0.990000 2.678033   
100 1.000000 2.704814   

x(1)=2.704814