I currently stumbled upon a time-dependent harmonic oscillator by trying to find a closed form solution for something like
$ \dot{\Theta}_{t}=-\Theta_{t}^{2}+\Lambda_{t}^{-1}$
where $\Theta$ and $\Lambda$ are $d\times d$-dimensional matrices and $\Lambda$ is symmetric and positive definite for $t\geq 0$, thus regular. I found some theorem connecting those Riccati differential equations with a linear system of differential equations. Namely "Radon's Lemma":
Let $M_{21}\left(t\right)\in\mathbb{R}^{m\times n}$, $M_{22}\left(t\right)\in\mathbb{R}^{m\times m}$, $M_{11}\left(t\right)\in\mathbb{R}^{n\times n}$, $M_{12}\left(t\right)\in\mathbb{R}^{n\times m}$ be locally integrable piecewise continuous matrix-valued functions for $t\in\left[t_{0},t_{f}\right]\subset\mathbb{R}$, we define matrix Riccati differential equations (RDE) as $ \dot{W}=M_{21}\left(t\right)+M_{22}\left(t\right)W-WM_{11}\left(t\right)-WM_{12}\left(t\right)W,\quad t\in\left[t_{0},t_{f}\right],$ then the following holds: If $Y=(\begin{array}{cc}Y_{1} & Y_{2}\end{array})^{\mathsf{T}}$ is a real solution of the linear system $\dot{Y}=MY\triangleq\left(\begin{array}{cc} M_{11} & M_{12}\\ M_{21} & M_{22} \end{array}\right)Y,\quad Y\left(t_{0}\right)=\left(\begin{array}{c} I_{n}\\ W_{0} \end{array}\right),$ such that $Y_{1}\left(t\right)\in\mathbb{R}^{n\times n}$ is regular for $t\in\left[t_{0},t_{f}\right]$, then $W:\left[t_{0},t_{f}\right]\rightarrow\mathbb{R}^{m\times n},\quad t\mapsto W\left(t\right)=Y_{2}\left(t\right)Y_{1}^{-1}\left(t\right)$ is a real solution of the RDE above.
So applying this to my problem, I have to solve
$ \left(\begin{array}{c} \dot{Y}_{t}^{\left(1\right)}\\ \dot{Y}_{t}^{\left(2\right)} \end{array}\right)=\left(\begin{array}{cc} 0 & I_{d}\\ \Lambda^{-1}_{t} & 0 \end{array}\right)\left(\begin{array}{c} Y_{t}^{\left(1\right)}\\ Y_{t}^{\left(2\right)} \end{array}\right),\quad Y_{T}^{\left(1\right)}=I_{d},Y_{T}^{\left(2\right)}=\mathbf{0}_{d}$
hence,
$ \dot{Y}_{t}^{\left(1\right)}=Y_{t}^{\left(2\right)}\\\dot{Y}_{t}^{\left(2\right)}=\Lambda^{-1}_{t}Y_{t}^{\left(1\right)}$
Differentiating the first and replacing using the second yields
$ 0=\ddot{Y}_{t}^{\left(1\right)}-\Lambda^{-1}_{t}Y_{t}^{\left(1\right)},$
This is the so called time-dependent harmonic oscillator and I have no clue how to solve this. More specifically, I want to know if there is a closed form solution to this and if not, whether I can say anything about regularity of some numerical solution to be able to apply Radon's Lemma.
Cheers!
The equation \begin{align*} \dot Y_t = A_t Y_t \end{align*} with $A_t=\begin{pmatrix}\mathbf0&\mathbf{1}\\\Lambda_t^{-1}& \mathbf0\end{pmatrix}$ is a linear time-depending ode. The initial value problem with given $Y_T$ has the solution \begin{align*} Y_t = \exp\left(\int_{\tau=T}^t A_\tau d\tau \right)Y_T. \end{align*}
Note, that if $\Lambda_t$ is positive definite then $\Lambda_t^{-1}$ is also positive definite. The eigenvalues of \begin{align*} A := \begin{pmatrix} \mathbf{0} & \mathbf{1}\\ \Lambda_t^{-1} & \mathbf{0} \end{pmatrix} \end{align*} are on the imaginary axis. You can employ BDF methods for such systems but they will artificially damp the solution. The modified extented version MEBDF is better. You can also try the trapezoidal rule that keeps imaginary eigenvalues of the continuous system on the imaginary axis for the discretized system. But, you need appropriate step-size control.