Let's **break down part (a)** into **2 steps** as requested, with explanations after each step and the final answer at the end. --- ### **Step 1: Write the ODE in Standard Form and Identify the Function** The given ODE is: \[ y' + 2y = x + 4,\quad y(0) = 1 \] Rearrange for standard form: \[ y' = x + 4 - 2y \] Let \( f(x, y) = x + 4 - 2y \). We use the **second-order Runge-Kutta (RK2, Heun's method)** with step size \( h = 0.2 \), initial value \( x_0 = 0, y_0 = 1 \). --- #### **Explanation after Step 1:** We have rewritten the ODE for use with RK2 and identified our function \( f(x, y) \) for numerical computation. --- ### **Step 2: Apply RK2 (Heun's Method) to Find \( y(0.2) \) and \( y(0.4) \)** **Heun’s method:** Given \( y_{n+1} = y_n + \frac{h}{2}(k_1 + k_2) \), where: - \( k_1 = f(x_n, y_n) \) - \( k_2 = f(x_n + h, y_n + h k_1) \) #### **First step: \( x_0 = 0, y_0 = 1 \), \( h = 0.2 \)** - \( k_1 = f(0, 1) = 0 + 4 - 2 \times 1 = 2 \) - \( y^* = y_0 + h k_1 = 1 + 0.2 \times 2 = 1.4 \) - \( k_2 = f(0.2, 1.4) = 0.2 + 4 - 2 \times 1.4 = 0.2 + 4 - 2.8 = 1.4 \) - \( y_1 = y_0 + \frac{0.2}{2}(2 + 1.4) = 1 + 0.1 \times 3.4 = 1 + 0.34 = 1.34 \) So, \( y(0.2) \approx 1.34 \) #### **Second step: \( x_1 = 0.2, y_1 = 1.34 \)** - \( k_1 = f(0.2, 1.34) = 0.2 + 4 - 2 \times 1.34 = 0.2 + 4 - 2.68 = 1.52 \) - \( y^* = y_1 + h k_1 = 1.34 + 0.2 \times 1.52 = 1.34 + 0.304 = 1.644 \) - \( k_2 = f(0.4, 1.644) = 0.4 + 4 - 2 \times 1.644 = 0.4 + 4 - 3.288 = 1.112 \) - \( y_2 = y_1 + \frac{0.2}{2}(1.52 + 1.112) = 1.34 + 0.1 \times 2.632 = 1.34 + 0.2632 = 1.6032 \) So, \( y(0.4) \approx 1.6032 \) --- #### **Explanation after Step 2:** We used Heun's method (RK2) for two steps to estimate \( y \) at \( x=0.2 \) and \( x=0.4 \). --- ## **Final Answer:** \[ \boxed{y(0.4) \approx 1.60} \] (Rounded to three significant digits)

Let's approach this step-by-step, as per your instructions. --- ### **Part (a): Approximate \( y(0.4) \) using RK2 with \( h=0.2 \) and initial condition \( y(0)=1 \).** --- ### **Step 1: Write the differential equation in standard form** Given: \[ y' + 2 y = x + 4, \quad y(0) = 1 \] Rearranged: \[ y' = x + 4 - 2 y \] Define: \[ f(x, y) = x + 4 - 2 y \] *Explanation:* We need this form to apply the Runge-Kutta method. --- ### **Step 2: Apply RK2 (Heun's method) to compute \( y \) at \( x=0.2 \) and \( x=0.4 \)** **Parameters:** - \( h = 0.2 \) - Initial point: \( x_0=0, y_0=1 \) --- ### **First step: From \( x=0 \) to \( x=0.2 \)** Calculate: - \( k_1 = f(x_0, y_0) = f(0, 1) = 0 + 4 - 2 \times 1 = 2 \) Estimate: - \( y^* = y_0 + h \times k_1 = 1 + 0.2 \times 2 = 1 + 0.4 = 1.4 \) Calculate: - \( k_2 = f(x_0 + h, y^*) = f(0.2, 1.4) = 0.2 + 4 - 2 \times 1.4 = 0.2 + 4 - 2.8 = 1.4 \) Update: \[ y_1 = y_0 + \frac{h}{2} (k_1 + k_2) = 1 + 0.1 \times (2 + 1.4) = 1 + 0.1 \times 3.4 = 1 + 0.34 = \boxed{1.34} \] --- ### **Second step: From \( x=0.2 \) to \( x=0.4 \)** Calculate: - \( k_1 = f(0.2, 1.34) = 0.2 + 4 - 2 \times 1.34 = 0.2 + 4 - 2.68 = 1.52 \) Estimate: - \( y^* = y_1 + h \times k_1 = 1.34 + 0.2 \times 1.52 = 1.34 + 0.304 = 1.644 \) Calculate: - \( k_2 = f(0.4, 1.644) = 0.4 + 4 - 2 \times 1.644 = 0.4 + 4 - 3.288 = 1.112 \) Update: \[ y_2 = y_1 + \frac{h}{2} (k_1 + k_2) = 1.34 + 0.1 \times (1.52 + 1.112) = 1.34 + 0.1 \times 2.632 = 1.34 + 0.2632 = \boxed{1.6032} \] --- ### **### Final Result:** \[ \boxed{ \textbf{Approximate } y(0.4) \approx \mathbf{1.60} } \] (Rounded to 2 decimal places) --- **Summary:** - **Step 1:** Derived the standard form and identified \( f(x,y) \). - **Step 2:** Applied RK2 (Heun’s method) in two steps to approximate \( y(0.4) \) as **1.60**.