1. **Problem:** Use Euler's method to solve the differential equation $$\frac{dy}{dx} = 1 - y$$ with initial condition $$y(0) = 0$$, step size $$h = 0.1$$, over the interval $$[0, 0.5]$$. 2. **Formula and explanation:** Euler's method approximates the solution of an ODE by the iterative formula: $$ y_{n+1} = y_n + h f(x_n, y_n) $$ where $$f(x,y)$$ is the right-hand side of the differential equation $$\frac{dy}{dx} = f(x,y)$$. 3. **Step-by-step calculation:** - Initial values: $$x_0 = 0$$, $$y_0 = 0$$. - Compute successive values for $$x_n$$ and $$y_n$$ using: $$y_{n+1} = y_n + h(1 - y_n)$$ Calculate each step: - Step 1: $$y_1 = y_0 + 0.1(1 - y_0) = 0 + 0.1(1 - 0) = 0.1$$ - Step 2: $$y_2 = y_1 + 0.1(1 - y_1) = 0.1 + 0.1(1 - 0.1) = 0.1 + 0.1 \times 0.9 = 0.19$$ - Step 3: $$y_3 = y_2 + 0.1(1 - y_2) = 0.19 + 0.1(1 - 0.19) = 0.19 + 0.1 \times 0.81 = 0.271$$ - Step 4: $$y_4 = y_3 + 0.1(1 - y_3) = 0.271 + 0.1(1 - 0.271) = 0.271 + 0.1 \times 0.729 = 0.3439$$ - Step 5: $$y_5 = y_4 + 0.1(1 - y_4) = 0.3439 + 0.1(1 - 0.3439) = 0.3439 + 0.1 \times 0.6561 = 0.40951$$ 4. **Result:** The approximate values of $$y$$ at $$x = 0, 0.1, 0.2, 0.3, 0.4, 0.5$$ are: $$ \begin{aligned} &y(0) = 0 \\ &y(0.1) = 0.1 \\ &y(0.2) = 0.19 \\ &y(0.3) = 0.271 \\ &y(0.4) = 0.3439 \\ &y(0.5) = 0.40951 \end{aligned} $$ This completes the Euler's method solution for the first problem.