You are using the Forward Euler method to solve the following initial value problem

𝑑
𝑦
𝑑
𝑥
=
𝑥
2
+
𝑦
4
𝑥
+
1

,

𝑦
(
0
)
=
1

A step of 1 is chosen. What is the value of the solution after one step?

𝑦
(
1
)
=
?

Question

You are using the Forward Euler method to solve the following initial value problem

𝑑
𝑦
𝑑
𝑥
=
𝑥
2
+
𝑦
4
𝑥
+
1
 
,
 
 
 
 
 
𝑦
(
0
)
=
1

A step of 1 is chosen. What is the value of the solution after one step?

𝑦
(
1
)
=
?

BlackTom AI · Accepted Answer

The problem asks us to apply the Forward Euler method to the IVP dy/dx = x^2 + y^4 x + 1 with y(0) = 1 and step size h = 1, and to find y(1).

First, identify the function f(x, y) on the right-hand side: f(x, y) = x^2 + y^4 x + 1.

Using the initial condition x0 = 0 and y0 = 1, compute the slope at the initial point: f(x0, y0) = f(0, 1) = (0)^2 + (1)^4 * (0) + 1 = 0 + 0 + 1 = 1.

Apply the Forward Euler update formula: y_{n+1} = y_n + h * f(x_n, y_n).
With h = 1, we get y(1) ≈ y(0) + 1 * f(0, 1) = 1 + 1 * 1 = 2.

Therefore, the Euler step yields y(1) = 2.

Note: If the interpretation of the term involving y were different (for example, a division by 4x, i.e., y/(4x)), the slope at x = 0 would be undefined due to division by zero, which would change the calculation entirely. The given form y^4 x allows a well-defined slope at x = 0 and leads to the value 2 for y(1).

You are using the Forward Euler method to solve the following initial value problem 𝑑 𝑦 𝑑 𝑥 = 𝑥 2 + 𝑦 4 𝑥 + 1 , 𝑦 ( 0 ) = 1 A step of 1 is chosen. What is the value of the solution after one step? 𝑦 ( 1 ) = ?单项选择题

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