To solve this problem using Euler's method, we proceed as follows:

Given:

1. Differential equation:
   $\frac{dy}{dx} = x^2 + y$
2. Initial condition:
   $y(0) = 0$
3. Step size: $h = 0.1$

We aim to approximate $y(0.4)$.

Euler's Method Formula:

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$ where $f(x, y) = x^2 + y$.

Iterative Steps:

Step 1: Initial values

$x_0 = 0, \quad y_0 = 0$

Step 2: Iteration for $x_1 = x_0 + h$, $y_1 = y_0 + h \cdot f(x_0, y_0)$

$x_1 = 0.1, \quad y_1 = 0 + 0.1 \cdot (0^2 + 0) = 0$

Step 3: Iteration for $x_2 = x_1 + h$, $y_2 = y_1 + h \cdot f(x_1, y_1)$

$x_2 = 0.2, \quad y_2 = y_1 + 0.1 \cdot (0.1^2 + 0) = 0 + 0.1 \cdot 0.01 = 0.001$

Step 4: Iteration for $x_3 = x_2 + h$, $y_3 = y_2 + h \cdot f(x_2, y_2)$

$x_3 = 0.3, \quad y_3 = y_2 + 0.1 \cdot (0.2^2 + 0.001) = 0.001 + 0.1 \cdot (0.04 + 0.001) = 0.001 + 0.0041 = 0.0051$

Step 5: Iteration for $x_4 = x_3 + h$, $y_4 = y_3 + h \cdot f(x_3, y_3)$

$x_4 = 0.4, \quad y_4 = y_3 + 0.1 \cdot (0.3^2 + 0.0051) = 0.0051 + 0.1 \cdot (0.09 + 0.0051) = 0.0051 + 0.00951 = 0.01461$

Approximate Solution:

$y(0.4) \approx 0.01461$

Let me know if you want a detailed table or further clarification.

Follow-up Questions:

1. What is the general idea behind Euler's method?
2. How does the choice of step size ($h$) affect the accuracy of the solution?
3. Can we improve this solution by using a smaller step size?
4. How does Euler's method compare to other numerical methods like Runge-Kutta?
5. What are the potential sources of error in numerical approximations?

Tip:

Always check your results with smaller step sizes to ensure the stability and accuracy of the numerical method.