Let's solve the problem in **2 steps** as requested: --- ### **Step 1: Convert the Second Order ODE to First Order System** Given: \[ \frac{d^2y}{dt^2} + 9y = 0 \] Let \( y_1 = y \) and \( y_2 = \frac{dy}{dt} \). Then: \[ \frac{dy_1}{dt} = y_2 \] \[ \frac{dy_2}{dt} = -9y_1 \] So, the system becomes: \[ \begin{cases} \frac{dy_1}{dt} = y_2 \\ \frac{dy_2}{dt} = -9y_1 \end{cases} \] With initial conditions: - \( y_1(0) = 1 \) - \( y_2(0) = 0 \) **Explanation:** This step rewrites the second-order ODE as a system of two first-order ODEs, which is necessary for both Euler's and RK4 methods. --- ### **Step 2: Numerical Solution Using Euler's and RK4 Methods** #### **(a) Euler's Method** Let step size \( h = 0.1 \). Euler's update formulas: \[ y_{1,n+1} = y_{1,n} + h \cdot y_{2,n} \] \[ y_{2,n+1} = y_{2,n} + h \cdot (-9y_{1,n}) \] #### **(b) RK4 Method** RK4 updates for system: For each step, compute the increments: \[ k_{1,1} = h f_1(t_n, y_{1,n}, y_{2,n}) = h y_{2,n} \] \[ k_{1,2} = h f_2(t_n, y_{1,n}, y_{2,n}) = h (-9 y_{1,n}) \] \[ k_{2,1} = h f_1(t_n + \frac{h}{2}, y_{1,n} + \frac{k_{1,1}}{2}, y_{2,n} + \frac{k_{1,2}}{2}) \] \[ k_{2,2} = h f_2(t_n + \frac{h}{2}, y_{1,n} + \frac{k_{1,1}}{2}, y_{2,n} + \frac{k_{1,2}}{2}) \] \[ k_{3,1} = h f_1(t_n + \frac{h}{2}, y_{1,n} + \frac{k_{2,1}}{2}, y_{2,n} + \frac{k_{2,2}}{2}) \] \[ k_{3,2} = h f_2(t_n + \frac{h}{2}, y_{1,n} + \frac{k_{2,1}}{2}, y_{2,n} + \frac{k_{2,2}}{2}) \] \[ k_{4,1} = h f_1(t_n + h, y_{1,n} + k_{3,1}, y_{2,n} + k_{3,2}) \] \[ k_{4,2} = h f_2(t_n + h, y_{1,n} + k_{3,1}, y_{2,n} + k_{3,2}) \] Then: \[ y_{1,n+1} = y_{1,n} + \frac{1}{6}(k_{1,1} + 2k_{2,1} + 2k_{3,1} + k_{4,1}) \] \[ y_{2,n+1} = y_{2,n} + \frac{1}{6}(k_{1,2} + 2k_{2,2} + 2k_{3,2} + k_{4,2}) \] **Explanation:** This step gives the explicit update rules for both numerical methods, ready for implementation with the given step size and interval. --- ### **Final Answer:** - Use the above update formulas to numerically solve for \( y \) from \( t=0 \) to \( t=4 \), with \( y(0)=1 \), \( y'(0)=0 \), and step size \( h=0.1 \). - Plot the results for both methods and compare with the exact solution \( y(t) = \cos(3t) \). --- **Summary:** 1. **Converted the second-order ODE to a first-order system.** 2. **Outlined Euler's and RK4 methods to solve the system numerically, ready for coding and plotting.** Let me know if you want the Python code for implementation and plotting!