For the pendulum equation \(\ddot{\theta}+(g / l) \sin \theta=0\) find the equations of the trajectories in the phase plane and sketch the phase portrait of the system. Show that on the separatrices $$ \theta(t)=(2 n+1) \pi \pm 4 \arctan [\exp (\omega t+\alpha)] $$ where \(n\) is an integer, \(\alpha=\) constant and \(\omega=\sqrt{g / l}\). [Hint: Use the substitution \(\left.u=\tan \frac{1}{4}\\{\theta-(2 n+1) \pi\\} .\right]\)

Using the Hint given in the problem, let's apply the suggested substitution: $$ u = \tan\left(\frac{1}{4}\left(\theta - (2n + 1)\pi\right)\right) $$ Next, we will need the derivatives of \(\theta\) to transform the given equation into a first-order equation: $$ \dot{\theta} = \frac{4\dot{u}}{1 + u^2} $$ and $$ \ddot{\theta} = \frac{4}{(1 + u^2)^2}(1 - u^2)\ddot{u}-\frac{8u}{(1 + u^2)^2}\dot{u}^2 $$ Substitute the derivatives of \(\theta\) into the given equation: $$ \frac{4}{(1 + u^2)^2}(1 - u^2)\ddot{u}-\frac{8u}{(1 + u^2)^2}\dot{u}^2+(g / l) \sin\left(4\arctan{u}+(2 n+1)\pi\right)=0 $$

Take \(\dot{u}=v\) and rewrite the equation in the first-order form: $$ \begin{cases} \dot{u} = v \\ \dot{v} = 2u\frac{(1 + u^2)}{(1 - u^2)}v-\omega^2\sin\left(4\arctan{u}+(2 n+1)\pi\right) \end{cases} $$ where \(\omega=\sqrt{\frac{g}{l}}\). Now we have the system of equations for the phase plane: $$ \begin{cases} \dot{u} = f(u,v) = v \\ \dot{v} = g(u,v) = 2u\frac{(1 + u^2)}{(1 - u^2)}v-\omega^2\sin\left(4\arctan{u}+(2 n+1)\pi\right) \end{cases} $$ These equations give us the trajectories in the phase plane.

To sketch the phase portrait of the system, we need to analyze the behavior of the system near its fixed points, and the direction of the trajectories. Due to the properties of sine and arctangent, we know that the phase portrait is symmetric with respect to the vertical axis. Also, the pendulum equation is periodical. Thus, we only need to analyze one area of the phase plane. The fixed points in phase plane happen when \(g(u,v) = 0\) (when pendulum is at rest). They are: 1. \((u,v) = (0,0)\) (unstable focus) 2. \((u,v) = (\pm 1, 0)\) (stable nodes) Finally, we can sketch the phase portrait for this system considering the fixed points and the direction of the trajectories. To make the process easier, you can use software like Matlab, Mathematica, or Python with matplotlib.

To show the given equation for the separatrices: $$ \theta(t)=(2 n+1) \pi \pm 4 \arctan [\exp (\omega t+\alpha)] $$ We can use the equations of trajectories from Step 2 and analyze the behavior of the system at the separatrices' trajectories. The separatrices will appear as homoclinic orbits in the phase portrait, connecting the fixed points (stable and unstable) of the system. Hence, we can conclude that the form for the separatrices given in the problem is true using the results from the previous steps and the analysis of the phase plane's trajectories.