Let's break down the steps needed to solve **Problem 2**: --- ### **Step 1: Write the Linearized Equation** The linearized equation given is: \[ \theta'' = -\frac{g}{l}\theta \] Given: - \(g = 9.81 \, \text{m/s}^2\) - \(l = 0.6 \, \text{m}\) - Initial conditions: \(\theta(0) = 10^\circ = 0.1745 \, \text{rad}\), \(\theta'(0) = 0\) \[ \frac{g}{l} = \frac{9.81}{0.6} = 16.35 \] So, \[ \theta'' + 16.35\theta = 0 \] --- ### **Step 2: Rewrite as First-Order System** Let \(y_1 = \theta\), \(y_2 = \theta'\): \[ \begin{cases} y_1' = y_2 \\ y_2' = -16.35y_1 \end{cases} \] with \(y_1(0) = 0.1745\), \(y_2(0) = 0\). --- ### **Step 3: Exact Solution** The general solution is: \[ \theta(t) = A\cos(\omega t) + B\sin(\omega t) \] where \(\omega = \sqrt{16.35} = 4.042\). Using initial conditions: - \(A = 0.1745\) - \(B = 0\) (since \(\theta'(0) = 0\)) So, \[ \theta_{\text{exact}}(t) = 0.1745 \cos(4.042 t) \] --- ### **Step 4: Numerical Schemes** For each time step (\(\Delta t = 0.15, 0.5, 1\)), solve from \(t=0\) to \(t=6\) seconds. #### (i) **Euler Method (Forward Euler)** \[ \begin{cases} y_1^{n+1} = y_1^n + \Delta t\, y_2^n \\ y_2^{n+1} = y_2^n - \Delta t\, 16.35 y_1^n \end{cases} \] #### (ii) **Backward Euler** \[ \begin{cases} y_1^{n+1} = y_1^n + \Delta t\, y_2^{n+1} \\ y_2^{n+1} = y_2^n - \Delta t\, 16.35 y_1^{n+1} \end{cases} \] (Solve as a system for \(y_1^{n+1}\), \(y_2^{n+1}\)) #### (iii) **Second-order Runge-Kutta (RK2, Midpoint)** \[ k_1 = f(y^n, t^n)\\ k_2 = f(y^n + \frac{\Delta t}{2}k_1, t^n + \frac{\Delta t}{2})\\ y^{n+1} = y^n + \Delta t \cdot k_2 \] #### (iv) **Fourth-order Runge-Kutta (RK4)** Standard RK4 steps for the 2D system. #### (v) **Trapezoidal Method (Crank-Nicolson)** \[ y^{n+1} = y^n + \frac{\Delta t}{2}(f(y^n, t^n) + f(y^{n+1}, t^{n+1})) \] (Again, requires solving a linear system at each step.) --- ### **Step 5: Implementation (Pseudocode)** For each method, initialize: - \(y_1^0 = 0.1745\) - \(y_2^0 = 0\) - Step through \(t_n = n\Delta t\), update using the method's formula. --- ### **Step 6: Discussion** #### **Accuracy and Stability** - **Euler**: Simple but conditionally stable. For oscillatory problems, it can become unstable for large \(\Delta t\). - **Backward Euler**: More stable (unconditionally for linear systems) but introduces damping (not physical for undamped oscillator). - **RK2**: Better accuracy, but still conditionally stable. - **RK4**: High accuracy and stability; preferred for oscillatory problems. - **Trapezoidal**: Unconditionally stable for linear problems and preserves oscillatory nature better than Backward Euler. #### **Expected Results** - For small \(\Delta t\), all methods should closely match the exact solution. - For larger \(\Delta t\), Forward Euler and RK2 may diverge or dampen; Backward Euler will dampen; Trapezoidal and RK4 will remain accurate and stable. --- ### **Step 7: Plots** Plot \(\theta(t)\) for each method and \(\Delta t\), and compare against the exact solution. --- ## **Summary Table** | Method | Stable for Large \(\Delta t\) | Preserves Oscillation | Accuracy | Complexity | |-----------------|-------------------------------|----------------------|----------|------------| | Euler | No | No | Low | Low | | Backward Euler | Yes | No (damps) | Low | Low | | RK2 | No | No | Medium | Medium | | RK4 | Yes (for this problem) | Yes | High | Higher | | Trapezoidal | Yes | Yes | High | Medium | --- ## **What to Submit** - Code for each method (if required) - Plots for each method and timestep - Discussion of accuracy and stability - Comparison with exact solution --- If you need example MATLAB/Python code or more detailed explanations for any method, let me know!