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The problem requires using Euler's Method with step size \( \Delta x = 0.1 \) to approximate the value of \( y \) when \( x = 1.3 \). The differential equation given is:

\[
\frac{dy}{dx} = x - 1
\]

with the initial condition \( y = 2 \) when \( x = 1 \).

### Euler's Method Steps:

Euler's Method updates the value of \( y \) based on the slope given by the differential equation. The iterative formula is:

\[
y_{\text{new}} = y_{\text{old}} + \Delta x \cdot f(x_{\text{old}}, y_{\text{old}})
\]

where \( f(x, y) = \frac{dy}{dx} = x - 1 \).

1. **Step 1: Initial condition**
   - \( x_0 = 1 \), \( y_0 = 2 \)
   - \( f(x_0, y_0) = 1 - 1 = 0 \)
   - Update: \( y_1 = y_0 + 0.1 \cdot 0 = 2 \)

2. **Step 2: \( x = 1.1 \)**
   - \( f(x_1, y_1) = 1.1 - 1 = 0.1 \)
   - Update: \( y_2 = y_1 + 0.1 \cdot 0.1 = 2 + 0.01 = 2.01 \)

3. **Step 3: \( x = 1.2 \)**
   - \( f(x_2, y_2) = 1.2 - 1 = 0.2 \)
   - Update: \( y_3 = y_2 + 0.1 \cdot 0.2 = 2.01 + 0.02 = 2.03 \)

4. **Step 4: \( x = 1.3 \)**
   - \( f(x_3, y_3) = 1.3 - 1 = 0.3 \)
   - Update: \( y_4 = y_3 + 0.1 \cdot 0.3 = 2.03 + 0.03 = 2.06 \)

### Final Answer:
The approximate value of \( y \) when \( x = 1.3 \) using Euler's Method is \( y \approx 2.06 \).

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Would you like further clarification or more details on Euler's Method?

Here are 5 related questions to expand on this topic:
1. How does the accuracy of Euler's Method change with different step sizes?
2. What are some limitations of Euler's Method for approximating solutions to differential equations?
3. Can you explain how Euler's Method relates to the concept of tangent line approximation?
4. How would you apply Euler's Method if the differential equation were nonlinear?
5. What are alternative numerical methods for solving differential equations, and how do they compare to Euler's Method?

**Tip:** Euler's Method is a first-order method, meaning the local error is proportional to the step size squared. Smaller step sizes typically result in more accurate approximations.

Use Euler's Method with increments of Δx = 0.1 to approximate the value of y when x = 1.3, given dy/dx = x - 1 and y = 2 when x = 1.

Learn how to approximate the value of y when x = 1.3 using Euler's Method with step size Δx = 0.1. The differential equation dy/dx = x - 1 is solved with an initial condition of y = 2 when x = 1. Step-by-step solution included.

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