Let $X(t)$ be a matrix-valued function which outputs symmetric and positive definite matrices for all $t$.
How can i solve
$$ \ddot{X}(t)= \alpha(t) X^{-1}(t). $$ where $\alpha(t)$ is some scalar. Additionally, i want to apply some initial conditions as $\dot{X}(0) =Y$ and $X(0)=X_0$.
I searched through literature and i tried a few things on my own, but i wasn't able to get something useful here. I also tried some ansatzes and solving for coefficients e.g.
$$ X(t) = X_0 \exp(t X_0^{-1} Y) $$ but i always arrived at dead ends.
Can someone give me a hint or point me to some literature? Is the first order equation easier: $\dot{X}(t)= \alpha(t) X^{-1}$?
Edit: Add time dependence of $\alpha$.