We will apply Euler's method to approximate $y(1.8)$ for the given differential equation:

$\frac{dy}{dx} = x^2 + y^2 - 2, \quad y(0) = 0$

Euler’s Method Formula:

Euler's method iteratively updates the solution using the formula:

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

where $f(x, y) = x^2 + y^2 - 2$, and $h$ is the step size.

We will calculate the approximations for $h = 0.1, 0.01, 0.001, 0.0001$, and compare results until two successive approximations agree up to two decimal places at $x = 1.8$.

Let me calculate the Euler approximations step by step.The results of applying Euler's method for different step sizes are:

• For $h = 0.1$, $y(1.8) \approx -0.50$
• For $h = 0.01$, $y(1.8) \approx -0.46$
• For $h = 0.001$, $y(1.8) \approx -0.45$
• For $h = 0.0001$, $y(1.8) \approx -0.45$

The approximate solution begins to stabilize between $h = 0.001$ and $h = 0.0001$, where the values agree when rounded to two decimal places.

Thus, a good approximation of $y(1.8)$ is:

$y(1.8) \approx -0.45$

Would you like more details on the steps or calculations? Here are five related questions:

1. How does Euler's method work conceptually for first-order differential equations?
2. What are the common sources of error in Euler's method?
3. How does reducing the step size in Euler's method improve accuracy?
4. How would the results change if a different initial condition were used?
5. Can other numerical methods, like the Runge-Kutta method, provide more accurate results?

Tip: Euler's method tends to accumulate error for larger step sizes; smaller steps improve accuracy but require more computation.