Let's solve **Question 1, part (a)** (Modified Euler's Method for two steps), and **part (c)** (compare to analytical solution), explaining each step: --- ## **Given:** \[ \frac{dy}{dx} = x^2 - 2y, \quad y(0) = \frac{1}{2}, \quad h = 0.2 \] Analytical solution: \( y = ce^{-2x} + \frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} \) --- ## **Step 1: Modified Euler’s Method (Heun’s Method)** ### **Initial values:** - \( x_0 = 0 \) - \( y_0 = 0.5 \) - \( h = 0.2 \) ### **First Step (from \( x_0 = 0 \) to \( x_1 = 0.2 \)):** 1. Compute slope at start: \( f(x_0, y_0) = x_0^2 - 2y_0 = 0^2 - 2 \times 0.5 = -1 \) 2. Predictor step: \( y^* = y_0 + h f(x_0, y_0) = 0.5 + 0.2 \times (-1) = 0.3 \) 3. Compute slope at end: \( f(x_1, y^*) = (0.2)^2 - 2(0.3) = 0.04 - 0.6 = -0.56 \) 4. Corrector step: \[ y_1 = y_0 + \frac{h}{2} \left[ f(x_0, y_0) + f(x_1, y^*) \right] = 0.5 + \frac{0.2}{2} (-1 - 0.56) = 0.5 + 0.1 \times (-1.56) = 0.5 - 0.156 = 0.344 \] **End of Step 1:** At \( x_1 = 0.2 \), \( y_1 \approx 0.344 \) --- ### **Second Step (from \( x_1 = 0.2 \) to \( x_2 = 0.4 \)):** 1. Compute slope at start: \( f(x_1, y_1) = (0.2)^2 - 2(0.344) = 0.04 - 0.688 = -0.648 \) 2. Predictor step: \( y^* = y_1 + h f(x_1, y_1) = 0.344 + 0.2 \times (-0.648) = 0.344 - 0.1296 = 0.2144 \) 3. Compute slope at end: \( f(x_2, y^*) = (0.4)^2 - 2(0.2144) = 0.16 - 0.4288 = -0.2688 \) 4. Corrector step: \[ y_2 = y_1 + \frac{h}{2} \left[ f(x_1, y_1) + f(x_2, y^*) \right] = 0.344 + 0.1 \times (-0.648 - 0.2688) = 0.344 + 0.1 \times (-0.9168) = 0.344 - 0.09168 = 0.25232 \] **End of Step 2:** At \( x_2 = 0.4 \), \( y_2 \approx 0.252 \) --- **Summary for Step 1 (Modified Euler):** - \( y_1 \approx 0.344 \) at \( x_1 = 0.2 \) - \( y_2 \approx 0.252 \) at \( x_2 = 0.4 \) --- ## **Step 2: Analytical Solution** Given: \[ y = ce^{-2x} + \frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} \] Use initial condition \( y(0) = \frac{1}{2} \): \[ \frac{1}{2} = c \cdot 1 + 0 - 0 + \frac{1}{4} \implies c = \frac{1}{2} - \frac{1}{4} = \frac{1}{4} \] So, \[ y(x) = \frac{1}{4}e^{-2x} + \frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} \] --- ### Evaluate at \( x = 0.2 \) and \( x = 0.4 \): #### At \( x = 0.2 \): \[ y(0.2) = \frac{1}{4}e^{-0.4} + \frac{0.04}{2} - \frac{0.2}{2} + \frac{1}{4} = \frac{1}{4}e^{-0.4} + 0.02 - 0.1 + 0.25 \] \[ e^{-0.4} \approx 0.6703 \] \[ y(0.2) = 0.25 \times 0.6703 + 0.02 - 0.1 + 0.25 = 0.1676 + 0.02 - 0.1 + 0.25 = 0.1676 + 0.17 = 0.3376 \] #### At \( x = 0.4 \): \[ y(0.4) = \frac{1}{4}e^{-0.8} + \frac{0.16}{2} - \frac{0.4}{2} + \frac{1}{4} = \frac{1}{4}e^{-0.8} + 0.08 - 0.2 + 0.25 \] \[ e^{-0.8} \approx 0.4493 \] \[ y(0.4) = 0.25 \times 0.4493 + 0.08 - 0.2 + 0.25 = 0.1123 + 0.08 - 0.2 + 0.25 = 0.1123 + 0.13 = 0.2423 \] --- ## **Final Answers:** ### **Modified Euler’s Method:** - \( y_1 \approx 0.344 \) at \( x_1 = 0.2 \) - \( y_2 \approx 0.252 \) at \( x_2 = 0.4 \) ### **Analytical Solution:** - \( y(0.2) \approx 0.338 \) - \( y(0.4) \approx 0.242 \) --- **Comparison:** The Modified Euler's Method is close to the analytical solution for these two steps. --- **If you need the Runge-Kutta (RK4) steps for part (b), let me know!**

The detailed solution I provided pertains to **Question 1**, specifically parts **(a)** (Modified Euler's Method) and **(c)** (comparison with the analytical solution). **To clarify:** - **Question 1** asks for: - **(a)** Using Modified Euler’s Method for two steps. - **(b)** Using the fourth-order Runge-Kutta (RK4) for two steps. - **(c)** Comparing the results to the analytical solution. - **(d)** Solving over the interval [0,1] with RK4 and exporting data to Excel. - **Question 2** involves solving the same differential equation over [0,1] using a 4th-order Runge-Kutta method (RK4), with code modifications to output data for comparison. --- ### **Final clarification:** **My previous response** covers **Question 1, parts (a) and (c)** — performing the Modified Euler approximation for two steps and comparing with the analytical solution. **Part (b)** (RK4 for two steps) and **Part (d)** (full interval solution with data export) are not included in that explanation. If you'd like, I can proceed to solve Part (b) with RK4 or help with Part (d) to prepare the code and data export. --- **Summary:** - Your question about **"Question 1.a,b,c and d"**: The solution provided is for **Question 1 parts (a) and (c)** only. - If you want the RK4 solution for part (b) or the full interval solution with data export for part (d), please specify!