Given the follow reaction data above, and using y(3)=17.8 and
Δ
𝑡
=3, use Euler's method to calculate y(9)= _____

Question

Given the follow reaction data above, and using y(3)=17.8 and 
Δ
𝑡
=3, use Euler's method to calculate y(9)= _____

BlackTom AI · Accepted Answer

First, identify what the table provides: at t = 3, 6, and 9, the entries under the row y = 17.8 are the slope values f(t, y) = dy/dt for that initial y. Specifically, starting from y(3) = 17.8, the slope at t = 3 is f = -2.5.

Step 1: Compute y at t = 6 using Euler's method with step h = Δt = 3.
- y(6) = y(3) + h * f(t = 3, y = 17.8) = 17.8 + 3 * (-2.5) = 17.8 - 7.5 = 10.3.

Step 2: Read the slope at t = 6 for y(6) from the table. The table shows f(t = 6, y = 17.8) = 3.5. This value represents dy/dt at t = 6 with the current y estimate, which Euler would use for the next step.

Step 3: Compute y at t = 9 using Euler's method with h = 3.
- y(9) = y(6) + h * f(t = 6, y = 17.8) = 10.3 + 3 * 3.5 = 10.3 + 10.5 = 20.8.

Thus, the Euler estimate for y at t = 9, given y(3) = 17.8 and Δt = 3, is 20.8. The value 20.8 is obtained by applying the two Euler steps first from t = 3 to t = 6, then from t = 6 to t = 9, using the corresponding slopes provided in the table.

If you compare with the derivative values in the table, you can verify: the first slope is -2.5 at t = 3, and the second slope is 3.5 at t = 6, which directly drive the successive updates to obtain y(6) and y(9) as shown above.

Given the follow reaction data above, and using y(3)=17.8 and Δ 𝑡 =3, use Euler's method to calculate y(9)= _____数值题

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