4. An initial value problem and its exact solution y(x) are given below. Apply Euler's method twice to approximate to this solution on the interval [O, ½], first with step size h 0.25 (two steps), then with step size h = 0.1 (five steps). Compare the three-decimal-place values of the two approximations at x = with the value y(%) of the actual solution. y' = (1 + y²),y(0) = 1; y(x) = tan (x+1) approx. "exact" Xn f(xn Yn) h f(xn Yn) Yn Yn+1 Xn+1 Yn+1 1 0.25 0.25 1.133519 1 0.25 0.25 0.5 1.287427 1 0.1 0.1 1.051293 1 00.1 0.1 0.2 1.105356 0.2 0.1 0.3 1.162492 0.3 0.1 0.4 1.223049 0.4 0.1 0.5 1.287427 3. 4-
4. An initial value problem and its exact solution y(x) are given below. Apply Euler's method twice to approximate to this solution on the interval [O, ½], first with step size h 0.25 (two steps), then with step size h = 0.1 (five steps). Compare the three-decimal-place values of the two approximations at x = with the value y(%) of the actual solution. y' = (1 + y²),y(0) = 1; y(x) = tan (x+1) approx. "exact" Xn f(xn Yn) h f(xn Yn) Yn Yn+1 Xn+1 Yn+1 1 0.25 0.25 1.133519 1 0.25 0.25 0.5 1.287427 1 0.1 0.1 1.051293 1 00.1 0.1 0.2 1.105356 0.2 0.1 0.3 1.162492 0.3 0.1 0.4 1.223049 0.4 0.1 0.5 1.287427 3. 4-
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![4. An initial value problem and its exact solution y(x) are given below. Apply Euler's method
twice to approximate to this solution on the interval [0, ½], first with step size h = 0.25
(two steps), then with step size h = 0.1 (five steps). Compare the three-decimal-place
values of the two approximations at x = h with the value y(%) of the actual solution.
y' = (1 + y?), y(0) = 1; y(x) = tan (x +1)
%3D
approx.
"ехact"
Xn
f(xn, Yn)
h f(xn Yn)
Yn
Yn+1
Xn+1
Yn+1
1
0.25
0.25
1.133519
1
0.25
0.25
0.5
1.287427
0 0
0.1
0.1
1.051293
1
00.1
0.1
0.2
1.105356
0.2
0.1
0.3
1.162492
3
0.3
0.1
0.4
1.223049
4
0.4
0.1
0.5
1.287427](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F404f3a78-f730-4773-ad7b-fe6612b342a7%2Fe801252e-da58-49f5-9816-0ad36cf66559%2Ftsxlpl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. An initial value problem and its exact solution y(x) are given below. Apply Euler's method
twice to approximate to this solution on the interval [0, ½], first with step size h = 0.25
(two steps), then with step size h = 0.1 (five steps). Compare the three-decimal-place
values of the two approximations at x = h with the value y(%) of the actual solution.
y' = (1 + y?), y(0) = 1; y(x) = tan (x +1)
%3D
approx.
"ехact"
Xn
f(xn, Yn)
h f(xn Yn)
Yn
Yn+1
Xn+1
Yn+1
1
0.25
0.25
1.133519
1
0.25
0.25
0.5
1.287427
0 0
0.1
0.1
1.051293
1
00.1
0.1
0.2
1.105356
0.2
0.1
0.3
1.162492
3
0.3
0.1
0.4
1.223049
4
0.4
0.1
0.5
1.287427
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