Let's solve each question step by step: --- ### **Q2. Gauss 3-point Integration Rule** Given: \[ I = \int_{-2}^{4} \frac{x}{\sqrt{2 + x^2}} dx \] #### **Step 1: Transform the interval from \([-2, 4]\) to \([-1, 1]\)** The general transformation is: \[ x = \frac{b-a}{2} t + \frac{a+b}{2} \] Here, \(a = -2\), \(b = 4\). \[ x = 3t + 1 \] \[ dx = 3 dt \] #### **Step 2: Transform the integral** \[ I = \int_{t=-1}^{t=1} \frac{3t+1}{\sqrt{2 + (3t+1)^2}} \cdot 3 dt = 3 \int_{-1}^{1} \frac{3t+1}{\sqrt{2 + (3t+1)^2}} dt \] #### **Step 3: 3-point Gauss-Legendre weights and abscissae** - \( t_1 = -\sqrt{\frac{3}{5}} \), \( t_2 = 0 \), \( t_3 = \sqrt{\frac{3}{5}} \) - \( w_1 = w_3 = \frac{5}{9} \), \( w_2 = \frac{8}{9} \) #### **Step 4: Evaluate the function at the nodes** Let \( f(t) = \frac{3t+1}{\sqrt{2 + (3t+1)^2}} \) - For \( t_1 = -\sqrt{\frac{3}{5}} \approx -0.7746 \): \[ x_1 = 3 \cdot (-0.7746) + 1 \approx -1.3238 \] \[ f(t_1) = \frac{-1.3238}{\sqrt{2 + (-1.3238)^2}} = \frac{-1.3238}{\sqrt{2 + 1.7535}} = \frac{-1.3238}{\sqrt{3.7535}} \approx \frac{-1.3238}{1.9373} \approx -0.6832 \] - For \( t_2 = 0 \): \[ x_2 = 3 \cdot 0 + 1 = 1 \] \[ f(t_2) = \frac{1}{\sqrt{2 + 1^2}} = \frac{1}{\sqrt{3}} \approx 0.5774 \] - For \( t_3 = 0.7746 \): \[ x_3 = 3 \cdot 0.7746 + 1 \approx 3.3238 \] \[ f(t_3) = \frac{3.3238}{\sqrt{2 + (3.3238)^2}} = \frac{3.3238}{\sqrt{2 + 11.0504}} = \frac{3.3238}{\sqrt{13.0504}} \approx \frac{3.3238}{3.6125} \approx 0.9197 \] #### **Step 5: Apply Gauss Quadrature formula** \[ I \approx 3 \left[ \frac{5}{9} \cdot f(t_1) + \frac{8}{9} \cdot f(t_2) + \frac{5}{9} \cdot f(t_3) \right] \] \[ I \approx 3 \left[ \frac{5}{9}(-0.6832) + \frac{8}{9}(0.5774) + \frac{5}{9}(0.9197) \right] \] Calculate each term: - \( \frac{5}{9}(-0.6832) \approx -0.3796 \) - \( \frac{8}{9}(0.5774) \approx 0.5137 \) - \( \frac{5}{9}(0.9197) \approx 0.5110 \) Sum: \[ -0.3796 + 0.5137 + 0.5110 = 0.6451 \] Multiply by 3: \[ I \approx 3 \times 0.6451 = 1.9353 \] **Final Answer for Q2:** \[ \boxed{1.94} \quad \text{(rounded to 2 decimal places)} \] --- ### **Q3. Modified Euler’s Method** Given: \[ \frac{dy}{dx} - 4x + \frac{3y}{x} = 0 \] or, \[ \frac{dy}{dx} = 4x - \frac{3y}{x} \] Initial: \(y(1) = 2\), step size \(h = 0.5\) We want \(y(1.5)\): #### **Step 1: First step, from \(x_0 = 1\) to \(x_1 = 1.5\)** Let \(f(x, y) = 4x - \frac{3y}{x}\). - \(x_0 = 1\), \(y_0 = 2\) **Predictor:** \[ f_0 = f(1, 2) = 4(1) - \frac{3 \times 2}{1} = 4 - 6 = -2 \] \[ y_1^{(p)} = y_0 + h f_0 = 2 + 0.5 \times (-2) = 2 - 1 = 1 \] **Corrector:** \[ f_1 = f(1.5, 1) = 4(1.5) - \frac{3 \times 1}{1.5} = 6 - 2 = 4 \] \[ y_1 = y_0 + \frac{h}{2} (f_0 + f_1) = 2 + 0.25 \times (-2 + 4) = 2 + 0.25 \times 2 = 2 + 0.5 = 2.5 \] **Final Answer for Q3:** \[ \boxed{y(1.5) = 2.5} \] --- ## **Summary of Answers** **Q2:** \( \boxed{1.94} \) **Q3:** \( \boxed{y(1.5) = 2.5} \) Let me know if you need further clarifications or additional steps!

Let's revisit **Question 3** carefully and perform the calculations step-by-step to ensure accuracy. --- ### **Q3. Solve using Modified Euler's Method** Given differential equation: \[ \frac{dy}{dx} = 4x - \frac{3y}{x} \] with initial condition: \[ y(1) = 2 \] Step size: \[ h = 0.5 \] We want to find \( y(1.5) \). --- ### **Step 1: Define the function \(f(x, y)\)** \[ f(x, y) = 4x - \frac{3y}{x} \] --- ### **Step 2: Calculate \(f(1, 2)\)** (initial point) \[ f(1, 2) = 4(1) - \frac{3 \times 2}{1} = 4 - 6 = -2 \] --- ### **Step 3: Predictor step** Estimate \(y\) at \(x=1.5\): \[ y_{predict} = y_0 + h \times f(x_0, y_0) = 2 + 0.5 \times (-2) = 2 - 1 = 1 \] --- ### **Step 4: Corrector step** Calculate \(f(1.5, y_{predict})\): \[ f(1.5, 1) = 4(1.5) - \frac{3 \times 1}{1.5} = 6 - 2 = 4 \] Now, compute the corrected value: \[ y_{1.5} = y_0 + \frac{h}{2} \left[ f(x_0, y_0) + f(x_1, y_{predict}) \right] \] \[ y_{1.5} = 2 + 0.25 \times (-2 + 4) = 2 + 0.25 \times 2 = 2 + 0.5 = 2.5 \] --- ### **Final result:** \[ \boxed{ \boxed{ y(1.5) = 2.5 } } \] --- ### **Summary:** The accurate calculation confirms that **\( y(1.5) = 2.5 \)** using the Modified Euler's Method with the given step size and initial condition. Please let me know if you'd like me to explain further or perform additional steps!