## Math Problem Statement

Consider the initial value problem given below. y primeequals1 minus y plus y cubed, y(0)equals0 Use the improved Euler's method with tolerance to approximate the solution to this initial value problem at xequals0.8. For a tolerance of epsilonequals0.003, use a stopping procedure based on the absolute error.

## Solution

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## Math Problem Analysis

### Mathematical Concepts

Numerical Methods

Differential Equations

Error Tolerance

Euler's Method

### Formulas

y_{n+1} = y_n + (h / 2) [f(x_n, y_n) + f(x_{n+1}, y_n + h f(x_n, y_n))]

f(x, y) = 1 - y + y^3

### Theorems

Improved Euler's Method (Heun's Method)

### Suitable Grade Level

Undergraduate level (Calculus / Numerical Analysis)

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