5.8. Numerical solution methods#
Solving \(y' = y + 5 \sin(2\pi t)\) using Euler’s Method
We are given the initial value problem:
Using the Euler method, we compute approximations for \(y(1)\) using two step sizes:
Using the Euler method, we compute approximations for \(y(1)\) using two step sizes:
Step size \(\Delta t = 0.1\)
Let \(f(t, y) = y + 5\sin(2\pi t)\)
\(n\) |
\(t_n\) |
\(y_n\) |
\(f(t_n, y_n)\) |
|---|---|---|---|
0 |
0.0 |
1.000 |
1.000 |
1 |
0.1 |
1.100 |
4.039 |
2 |
0.2 |
1.504 |
6.257 |
3 |
0.3 |
2.130 |
6.895 |
4 |
0.4 |
2.818 |
5.757 |
5 |
0.5 |
3.394 |
3.394 |
6 |
0.6 |
3.733 |
0.795 |
7 |
0.7 |
3.813 |
-0.942 |
8 |
0.8 |
3.719 |
-1.037 |
9 |
0.9 |
3.615 |
0.676 |
10 |
1.0 |
3.683 |
— |
So, for \(\Delta t = 0.1\), the Euler approximation gives:
Step size \(\Delta t = 0.2\)
Step size \(\Delta t = 0.2\)
\(n\) |
\(t_n\) |
\(y_n\) |
\(f(t_n, y_n)\) |
|---|---|---|---|
0 |
0.0 |
1.000 |
1.000 |
1 |
0.2 |
1.200 |
6.955 |
2 |
0.4 |
2.591 |
5.330 |
3 |
0.6 |
3.657 |
0.576 |
4 |
0.8 |
3.772 |
-1.155 |
5 |
1.0 |
3.322 |
— |
Therefore, for \(\Delta t = 0.2\):