Consider the matrix differential equation in PSD matrix $X \in \mathbb{R}^{N \times N}$
$$\dot X(t) = X(t) F(X(t))^T + F(X(t))X(t),$$
where $F(X):\mathbb{R}^{N \times N} \to \mathbb{R}^{N \times N}$ is a matrix-valued function.
I wonder if I can apply the below discretization, which is a mix of the explicit and implicit Euler method, and ensure the solution to be consistent
$$\frac{X_{k+1}-X_k}{\Delta t} = X_{k+1} F(X_k)^T + F(X_k)X_{k+1}$$