Consider the following IVP

𝑑
𝑦
𝑑
𝑥
=
𝑦
+
𝑥
,

𝑦
(
0
)
=
1

Use a step h=0.5. What is the solution at x=0.5 predicted by the Backward Euler method?

Question

Consider the following IVP

𝑑
𝑦
𝑑
𝑥
=
𝑦
+
𝑥
,
 
 
 
 
 
𝑦
(
0
)
=
1

Use a step h=0.5. What is the solution at x=0.5 predicted by the Backward Euler method?

BlackTom AI · Accepted Answer

We are given an initial value problem: dy/dx = y + x with y(0) = 1, and we are to use backward Euler with step size h = 0.5 to approximate the solution at x = 0.5.

First, identify the update formula for backward Euler: y_{n+1} = y_n + h * f(x_{n+1}, y_{n+1}), where f(x,y) = y + x.

Here, x_0 = 0, y_0 = 1, h = 0.5, so x_1 = x_0 + h = 0.5. The implicit equation becomes:
y_1 = y_0 + h * (y_1 + x_1).

Plug in the known values: y_1 = 1 + 0.5 * (y_1 + 0.5).

Distribute and collect like terms: y_1 = 1 + 0.5*y_1 + 0.25 => y_1 - 0.5*y_1 = 1.25.

This simplifies to 0.5*y_1 = 1.25, so y_1 = 1.25 / 0.5 = 2.5.

Thus, the backward Euler prediction for the solution at x = 0.5 is 2.5 when compared against the listed option set.

Option provided: 2.5

Consider the following IVP 𝑑 𝑦 𝑑 𝑥 = 𝑦 + 𝑥 , 𝑦 ( 0 ) = 1 Use a step h=0.5. What is the solution at x=0.5 predicted by the Backward Euler method?单项选择题

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