We are tasked with using the Euler method to approximate the solution to the initial value problem (IVP):

$y' = x + y^2, \quad y(0) = 0$

where $h = 0.1$, and we are asked to approximate $y(0.2)$. Euler's method updates the value of $y$ using the formula:

$y_{n+1} = y_n + h f(x_n, y_n)$

Here, $f(x, y) = x + y^2$, and we start with the initial condition $y(0) = 0$.

Step-by-step process:

1. Initial condition:
   $x_0 = 0$, $y_0 = 0$.

2. Use Euler's formula for $x_1 = 0.1$:

   We calculate $y_1$: $y_1 = y_0 + h \cdot f(x_0, y_0)$ Substituting the values: $f(x_0, y_0) = 0 + 0^2 = 0$ Therefore, $y_1 = 0 + 0.1 \cdot 0 = 0$

3. Now for $x_2 = 0.2$:

   We calculate $y_2$: $y_2 = y_1 + h \cdot f(x_1, y_1)$ Substituting the values: $f(x_1, y_1) = 0.1 + 0^2 = 0.1$ Therefore, $y_2 = 0 + 0.1 \cdot 0.1 = 0.01$

Final answer:

Using the Euler method with $h = 0.1$, we approximate $y(0.2) \approx 0.01$.

Would you like to explore further details on the method or how to improve the approximation with a smaller step size?

Here are some related questions:

1. What is the order of accuracy for the Euler method?
2. How does the step size $h$ influence the accuracy of the method?
3. What other numerical methods can be used for solving IVPs more accurately?
4. How can we derive the Euler method from the Taylor series expansion?
5. What would happen if we increased the step size to $h = 0.2$?

Tip: Smaller step sizes generally provide more accurate approximations in numerical methods like Euler's.