Let's **solve problem 2(a)** using the Euler method, as requested: --- ### Given: - Differential equation: \( y' = -y \) - Initial condition: \( y(0) = 1 \) - Step size: \( h = 0.2 \) We need to compute \( y_1, y_2, y_3 \) using Euler's method. --- ## **Step 1: Euler's Method Formula** Euler's method for \( y_{n+1} = y_n + h f(x_n, y_n) \) Here, \( f(x, y) = -y \). --- ## **Step 2: Set Initial Values** - \( x_0 = 0 \), \( y_0 = 1 \) --- ## **Step 3: Compute Each Step** ### **Step 1: Calculate \( y_1 \)** \[ x_1 = x_0 + h = 0 + 0.2 = 0.2 \] \[ y_1 = y_0 + h f(x_0, y_0) = 1 + 0.2 \cdot (-1) = 1 - 0.2 = 0.8 \] --- ### **Step 2: Calculate \( y_2 \)** \[ x_2 = x_1 + h = 0.2 + 0.2 = 0.4 \] \[ y_2 = y_1 + h f(x_1, y_1) = 0.8 + 0.2 \cdot (-0.8) = 0.8 - 0.16 = 0.64 \] --- ### **Step 3: Calculate \( y_3 \)** \[ x_3 = x_2 + h = 0.4 + 0.2 = 0.6 \] \[ y_3 = y_2 + h f(x_2, y_2) = 0.64 + 0.2 \cdot (-0.64) = 0.64 - 0.128 = 0.512 \] --- ## **Final Answers:** \[ \boxed{ y_1 = 0.8,\quad y_2 = 0.64,\quad y_3 = 0.512 } \] --- ### **Summary Table** | \( n \) | \( x_n \) | \( y_n \) | |:--------:|:---------:|:---------:| | 0 | 0.0 | 1.000 | | 1 | 0.2 | 0.800 | | 2 | 0.4 | 0.640 | | 3 | 0.6 | 0.512 | Let me know if you want solutions for the other parts or an explanation for the exact solution!