Problem 2.
Approximate the motion of simple pendulum in case of $\theta(0)=0$, $\theta'(0)=v$.
We have seen in Problem 1 the differential equation of the motion of the simple pendulum is
\[
\theta\,''(t)+\frac{g}{\ell}\sin(\theta(t))=0.
\]
Substituting $t:=0$ we obtain
\[
\theta\,''(0)+\frac{g}{\ell}\sin(\theta(0))=0.
\]
It follows
\[
\theta\,''(0)=0.
\]
Taking the derivative of the equation we have
\[
\theta\,'''(t)+\frac{g}{\ell}\cos(\theta(t))\theta\,'(t)=0.
\]
Substituting $t:=0$ we obtain
\[
\theta\,'''(0)+\frac{g}{\ell}\theta\,'(0)=0.
\]
It follows
\[
\theta\,'''(0)=-\frac{g}{\ell}v.
\]
Then we get
\begin{align*}
\theta(t) &=\theta\,(0)+\theta\,'(0)t+\frac{1}{2!}\theta\,''(0)t^2+\frac{1}{3!}\theta\,'''(0)t^3+\ldots\\
&=vt-\frac{g}{6\ell}vt^3+\ldots\qquad\blacksquare
\end{align*}
