1. (Euler’s method) First, work out the first three steps by hand. Then approximate y(2) for...

1. (Euler’s method) First, work out the first three steps by hand. Then approximate y(2) for each of the initial value problems using Euler’s method, first with a step size of h = .1 and then with a step size of h = .05 using the Excel spreadsheet. (a) dy dx = 2xy, y(0) = 1 (b) dy dx = x − y x + 2y , y(0) = 1 (c) dy dx = y + x, y(0) = 1 ignore the excell sheet portion

Expert Solution

			h =0.1

n	x_n	y_n
0	0	1
1	0.1	1
2	0.2	1.02
3	0.3	1.0608
4	0.4	1.124448
5	0.5	1.214404
6	0.6	1.335844
7	0.7	1.496146
8	0.8	1.705606
9	0.9	1.978503
10	1	2.334633
11	1.1	2.80156
12	1.2	3.417903
13	1.3	4.2382
14	1.4	5.340132
15	1.5	6.835369
16	1.6	8.88598
17	1.7	11.72949
18	1.8	15.71752
19	1.9	21.37583
20	2	29.49864

	h=0.05

n	x_n	y_n
0	0	1
1	0.05	1
2	0.1	1.005
3	0.15	1.01505
4	0.2	1.030276
5	0.25	1.050881
6	0.3	1.077153
7	0.35	1.109468
8	0.4	1.148299
9	0.45	1.194231
10	0.5	1.247972
11	0.55	1.31037
12	0.6	1.382441
13	0.65	1.465387
14	0.7	1.560637
15	0.75	1.669882
16	0.8	1.795123
17	0.85	1.938733
18	0.9	2.103525
19	0.95	2.292842
20	1	2.510662
21	1.05	2.761729
22	1.1	3.05171
23	1.15	3.387398
24	1.2	3.776949
25	1.25	4.230183
26	1.3	4.758956
27	1.35	5.37762
28	1.4	6.103599
29	1.45	6.958102
30	1.5	7.967027
31	1.55	9.162081
32	1.6	10.5822
33	1.65	12.27536
34	1.7	14.30079
35	1.75	16.73192
36	1.8	19.66001
37	1.85	23.19881
38	1.9	27.49059
39	1.95	32.71381
		40	2	39.093

part(b):


	h=0.1

n	x_n	y_n
0	0	1
1	0.1	1.2
2	0.2	1.438
3	0.3	1.71684
4	0.4	2.038703
5	0.5	2.404895
6	0.6	2.81563
7	0.7	3.269818
8	0.8	3.764894
9	0.9	4.296681
10	1	4.859316
11	1.1	5.445248
12	1.2	6.04532
13	1.3	6.648946
14	1.4	7.244372
15	1.5	7.819034
16	1.6	8.359986
17	1.7	8.854385
18	1.8	9.290017
19	1.9	9.655817
20	2	9.942375

	h=0.05

n	x_n	y_n
0	0	1
1	0.05	1.1
2	0.1	1.20975
3	0.15	1.329676
4	0.2	1.460171
5	0.25	1.601587
6	0.3	1.754226
7	0.35	1.918335
8	0.4	2.094097
9	0.45	2.281625
10	0.5	2.480951
11	0.55	2.692022
12	0.6	2.914694
13	0.65	3.148723
14	0.7	3.393761
15	0.75	3.649356
16	0.8	3.914941
17	0.85	4.189837
18	0.9	4.473253
19	0.95	4.764282
20	1	5.061906
21	1.05	5.365002
22	1.1	5.672339
23	1.15	5.982595
24	1.2	6.294355
25	1.25	6.606129
26	1.3	6.916359
27	1.35	7.223431
28	1.4	7.525693
29	1.45	7.821464
30	1.5	8.109054
31	1.55	8.38678
32	1.6	8.652983
33	1.65	8.906043
34	1.7	9.144398
35	1.75	9.366564
36	1.8	9.571146
37	1.85	9.756858
38	1.9	9.922534
39	1.95	10.06715
40	2	10.18981

part(c):

Colby Messinger answered 3 years ago

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