Let y(t) be the solution to y = 4te satisfying y(0) = 2. 0.2 to approximate y(0.2), y(0.4), ..., y(1.0). (a) Use Euler's Method with time step h = ktk 00 10.2 20.4 30.6 40.8 51.0 Yk 2 (b) Use separation of variables to find y(t) exactly. y(t) = (c) Compute the error in the approximations to y(0.2), y(0.6), and y(1). |y(0.2) - y₁|= |y(0.6) - y3|= |y(1) ― y5| =
Let y(t) be the solution to y = 4te satisfying y(0) = 2. 0.2 to approximate y(0.2), y(0.4), ..., y(1.0). (a) Use Euler's Method with time step h = ktk 00 10.2 20.4 30.6 40.8 51.0 Yk 2 (b) Use separation of variables to find y(t) exactly. y(t) = (c) Compute the error in the approximations to y(0.2), y(0.6), and y(1). |y(0.2) - y₁|= |y(0.6) - y3|= |y(1) ― y5| =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( y(t) \) be the solution to \( \dot{y} = 4te^{-y} \) satisfying \( y(0) = 2 \).
(a) Use Euler's Method with time step \( h = 0.2 \) to approximate \( y(0.2), y(0.4), \ldots, y(1.0) \).
\[
\begin{array}{|c|c|c|}
\hline
k & t_k & y_k \\
\hline
0 & 0 & 2 \\
\hline
1 & 0.2 & \\
\hline
2 & 0.4 & \\
\hline
3 & 0.6 & \\
\hline
4 & 0.8 & \\
\hline
5 & 1.0 & \\
\hline
\end{array}
\]
(b) Use separation of variables to find \( y(t) \) exactly.
\( y(t) = \)
(c) Compute the error in the approximations to \( y(0.2), y(0.6), \) and \( y(1) \).
\[
|y(0.2) - y_1| =
\]
\[
|y(0.6) - y_3| =
\]
\[
|y(1) - y_5| =
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f1c68a1-c113-41cc-b0cf-42f2666e5687%2F234a5795-a6b9-4dc7-8732-812e35885d5c%2F4t1clgl_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( y(t) \) be the solution to \( \dot{y} = 4te^{-y} \) satisfying \( y(0) = 2 \).
(a) Use Euler's Method with time step \( h = 0.2 \) to approximate \( y(0.2), y(0.4), \ldots, y(1.0) \).
\[
\begin{array}{|c|c|c|}
\hline
k & t_k & y_k \\
\hline
0 & 0 & 2 \\
\hline
1 & 0.2 & \\
\hline
2 & 0.4 & \\
\hline
3 & 0.6 & \\
\hline
4 & 0.8 & \\
\hline
5 & 1.0 & \\
\hline
\end{array}
\]
(b) Use separation of variables to find \( y(t) \) exactly.
\( y(t) = \)
(c) Compute the error in the approximations to \( y(0.2), y(0.6), \) and \( y(1) \).
\[
|y(0.2) - y_1| =
\]
\[
|y(0.6) - y_3| =
\]
\[
|y(1) - y_5| =
\]
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