I would like to find a closed-form, or determine that none exists, for the ODE $\frac{d\mathbf{x}}{dt} = \cos(\mathbf{c}^T \mathbf{x})\mathbf{a} - \sin(\mathbf{c}^T \mathbf{x})\mathbf{b},$ where $a,b,c,\mathbf{x} \in \mathbb{R}^d$. When $d=1$, the solution is given in closed form here.
My question is: does such a closed-form solution exist for $d>1$?
Partial Progress: When $b=0$, the problem reduces to $\frac{d\mathbf{x}}{dt} = \cos(\mathbf{c}^T \mathbf{x})\mathbf{a}.$
In this case we may instead take $y = \mathbf{c}^T\mathbf{x} \in \mathbb{R}$ and solve the differential equation $\frac{d\mathbf{y}}{dt} = \mathbf{c}^T\mathbf{a}\cos(y)$, which admits a [closed-form solution2. Exploiting the fact that $x_t - x_0 \propto \mathbf{a}$, we can use the closed form for $\mathbf{c}^Tx$ to solve for $x_t$. A similar approach can be taken when $\mathbf{a}=0$.