Key for estimations:
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|
|
|
|
| 0 | 5 | 2 | 2 | -3 | -3 |
| 1 | 7 | -1 | -1 | -2 | -2 |
| 2 | 6 | -3 | -3 | -1 | -1 |
| 3 | 3 | -4 | -4 | 0 | 0 |
| 4 | -1 | -4 | -4 | 1 | 1 |
The exact solution for this problem is 
and
y
= x 3/6 - 3/2 x2 + 2x + 5
.
Thus y(4)= 64/6 -48/2 +8 +5 = -2/3.
In fact y(3) = 27/6 -27/2 + 6 +5 = 2. Since y'' < 0 for x < 3, the solution function has a graph that is concave down for x <3, so the differential estimator used in Euler's method is an overestimate. However, the solution is concave up for 3 < x < 4,and so the estimate can change, as it does, to an underestimate despite the accumulated error in the estimation of y(3).
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| 0 | 0 | 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | -1 | -1 |
| 2 | 2 | 0 | 0 | -2 | -2 |
| 3 | 2 | -2 | -2 | -2 | -2 |
| 4 | 0 | -4 | -4 | 0 | 0 |
|
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| 0.00 | 5.00 | 2.00 | 1.00 | 5.00 | 2.50 |
| 0.50 | 6.00 | 4.50 | 2.25 | 6.00 | 3.00 |
| 1.00 | 8.25 | 7.50 | 3.75 | 8.25 | 4.13 |
| 1.50 | 12.00 | 11.63 | 5.81 | 12.00 | 6.00 |
| 2.00 | 17.81 | 17.63 | 8.81 | 17.81 | 8.91 |
| 2.50 | 26.63 | 26.53 | 13.27 | 26.63 | 13.31 |
| 3.00 | 39.89 | 39.84 | 19.92 | 39.89 | 19.95 |
| 3.50 | 59.81 | 59.79 | 29.89 | 59.81 | 29.91 |
| 4.00 | 89.71 | 89.70 | 44.85 | 89.71 | 44.85 |