Given the system of the following two ODEs for $x,y$ functions of an independent variable $t$
$$0=\frac{1}{2} \sigma^{2} x''+\left(\mu-\tilde{K} (\frac{x'}{2}+y')\right)\ x'-\delta x+\left(t-\rho\right)^2,\\ 0=\frac{1}{2} \sigma^{2} y''+\left(\mu-\tilde{K} (\frac{y'}{2}+x')\right)\ y'-\delta y+\left(t-\rho\right)^2$$
where the parameters $\sigma>0,\mu\in\mathbb{R},\tilde{K}\geq 0,\delta>0,\rho\in\mathbb{R}$, I have made the transformation into 4 ODEs of first order being
$$ 0=\frac{1}{2} \sigma^{2} \xi'+\left(\mu-\tilde{K} (\frac{\xi}{2}+\Psi)\right)\ \xi-\delta x+\left(t-\rho\right)^2,\\ 0=\frac{1}{2} \sigma^{2} \Psi'+\left(\mu-\tilde{K} (\frac{\Psi}{2}+\xi)\right)\ \Psi-\delta y+\left(t-\rho\right)^2,\\ \xi=x',\quad\Psi=y'. $$
As this system is strongly symmetric and every attempt at finding a general solution has failed, I am now trying to see if I can transform this into a matrix Riccati equation, i.e. I want to find matrices such that the system $$\mathbb{V}'=\mathbb{V}Q\mathbb{V}+S\mathbb{V}+T$$ describes the system of my 4 first-order ODEs.
The reason for this is that I have already found 4 particular solutions (polynomials of second order) and I want to specify how a general solution looks. I have the hope that for an algebraic Riccati equation I can reduce the system to one I can solve and receive a closed form solution.
However I am not sure
- how to choose the dimensions
- how to calculate the parameters
- if it even is possible...
I have tried $2\times 2$ but keep struggling with the $\mathbb{V}Q\mathbb{V}$-term...
Does trying higher dimensions even change the possible outcomes of this, i.e. is there a point in searching for a solution, where $\mathbb{V}$ is not of full rank?
Is there an approach I can use to calculate the matrices or do I just have to keep throwing random approaches at this? Any help / tips on approaching this problem are welcome!