This is the mathematical pendulum equation for the oscillatory case.
Mathematica solves it directly, but the form of the solution is a bit weird, because the Jacobi amplitude is primarily defined for rotational modes with veloctiy :>0 at [Theta] = Pi,.
DSolveValue[\[Theta]''[t] - \[Pi]^2/8 Sin[2 \[Theta][t]] == 0 , \[Theta][t], t]
JacobiAmplitude[(I Sqrt[\[Pi]^2-8 C[1] (t+C[2]))/(2 Sqrt[2]),(2 \[Pi]^2)/(\[Pi]^2-8 C[1])]
Its nearly impossible to include the start condtions. As the equation is time independent, one constant C[1] is simply the start time and has be set so to meet the start conditions together with .the second constant,determines the velocity at t=-C[2] , that is in the lower dead point.
D[JacobiAmplitude[(I Sqrt[\[Pi]^2 - 8 C[1]] (t + C[2]))/(2 Sqrt[2]), ( 2 \[Pi]^2)/(\[Pi]^2 - 8 C[1])], t] /. {t + C[2] -> 0}
\Omega = (I Sqrt[\[Pi]^2 - 8 C[1]])/(2 Sqrt[2]) -> (Sqrt[8 C[1]]-\[Pi]^2 )/(2 Sqrt[2])
We know that the total energy is conserved
[Theta]'[t] [Theta]''[t] - \[Pi]^2/8 [Theta]'[t] Sin[2 \[Theta][t]] == 0 -->
1/2 [Theta]'[t] ^2 +\[Pi]^2/16 ( 1- Cos[2 \Theta] = W = const
From this equation it is possible to solve for C[1] in terms of Theta
Because of the poor state of implementation of the elliptic function complex in Mathematica, most work has to be done by hand. The rather trival solution can be found in Pendel exakte Lösung