Use​ Euler's method to approximate ​y(1.8​). Start with step size hequals0.1​, and then use successively smaller step sizes ​(hequals0.01​, 0.001​, 0.0001​, ​etc.) until successive approximate solution values at xequals1.8 agree rounded off to two decimal places.
y prime equals x squared plus y squared minus 2​, y left parenthesis 0 right parenthesis equals 0
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Part 1
The approximate solution values at xequals1.8 begin to agree rounded off to two decimal places between 
▼ 
h equals 0.01 and h equals 0.001 .
h equals 0.001 and h equals 0.0001 .
h equals 0.1 and h equals 0.01 .
 ​So, a good approximation of ​y(1.8​) is 
  
enter your response here.
​(Type an integer or decimal rounded to two decimal places as​ needed.)

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.

Use Euler's method to approximate y(1.8). Start with step size h = 0.1, and then use successively smaller step sizes (h = 0.01, 0.001, 0.0001, etc.) until successive approximate solution values at x = 1.8 agree rounded off to two decimal places. y' = x^2 + y^2 - 2, y(0) = 0.

Learn how to use Euler's method to approximate y(1.8) for the differential equation y' = x^2 + y^2 - 2, with y(0) = 0. This example explores step sizes from 0.1 to 0.0001, illustrating the accuracy of numerical methods in solving differential equations.

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