I am stuck with a little problem that I cannot solve mith the standard methods I learn at university. I have a system of coupled ODEs:

$f'(t) = P \cos(k t + \Phi_1) g(t)$

$g'(t) = Q \cos(k t + \Phi_2) f(t)$

From what I read, the is no teaching book-way to solve such a system. I found question 66172 to be similar, but my values are $P,Q$ up to $\approx 10^3-10^4$ and $k$,$\Phi_{1,2}$ basically arbitrary ($k$ ranges from $-10^5$ to $10^5$ explicitly including 0). Only $t$ may be restricted to the order of magnitude around $10^{-2}$. Boundary conditions may be $f(0)=1$, $g(0)=0$ and nice to have would be $f(0) \approx g(0)$

I have done numerical calculations and for my numbers, the solutions do not look to crazy. For $k=0$ the system is solvable and basically gives oscillations between the two functions. For $k\neq0$ I basically geo Oszillations withs higher frequency and lower amplitude (e.g., $f(t)$ oszillating between 1 and 0.8 and $g(t)$ between 0 and 0.2). For $k\gg10^5$, the system becomes mostly stationary.

I would really appreciate suggestions for an analytical approach, at least as an approximation.