1. **Problem Statement:** Find the value of $y$ at $x=0.1$ using Euler's modified method with step size $h=0.1$ for the differential equation $$\frac{dy}{dx} = 1 - y, \quad y(0) = 0.$$ 2. **Formula and Explanation:** Euler's modified method (also called Heun's method) improves the basic Euler method by using the average slope over the interval: $$k_1 = f(x_n, y_n)$$ $$k_2 = f(x_n + h, y_n + h k_1)$$ $$y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2)$$ where $f(x,y) = \frac{dy}{dx}$. 3. **Step-by-step Calculation:** - Initial values: $x_0=0$, $y_0=0$, $h=0.1$ - Compute $k_1 = f(0,0) = 1 - 0 = 1$ - Compute $k_2 = f(0.1, 0 + 0.1 \times 1) = f(0.1, 0.1) = 1 - 0.1 = 0.9$ - Update $y_1 = 0 + \frac{0.1}{2}(1 + 0.9) = 0.05 \times 1.9 = 0.095$ 4. **Final answer:** $$y(0.1) \approx 0.095$$ This completes the solution for the first question.