I have the following differential equation, I am not sure how to integrate this.
$$\left((\ddot y - \ddot M y)\delta_{t,\pi/2} + (\dot y - \dot MM^{-1}y)(1-\delta_{t,\pi/2}) \right) = 0$$
Now when $t\ne\pi/2$, I know that the equation comes to $$\begin{align} &(\dot y - \dot MM^{-1}y) dt = 0\\ \end{align}$$ which has a solution $y = \exp(\int\dot MM^{-1} dt)y_0$ for $y_0 = y(0)$.
However, I don't know how to evaluate when $t=\pi/2$.
Any help is appreciated.
The integrating factor is $M^{-1}$, $$\frac{d}{dt}(M^{-1}y)=M^{-1}\dot y-M^{-1}\dot MM^{-1}y=0\implies y(t)=M(t)C.$$
What you wrote makes sense for scalar functions, with the exponential of matrices however you have to be careful, as the scalar treatment only generalizes if everything commutes.
If one understands the first term as a very unusual way to write an initial condition, one concludes from $\ddot M(\frac\pi2)C=\ddot y(\frac\pi2)=\ddot M(\frac\pi2)y(\frac\pi2)$ that $C=y(\frac\pi2)$ which in turn necessitates $$ (M(\tfrac\pi2)-I)y(\tfrac\pi2)=0 $$