How can I solve this system of equation analytically?

Question

$$ \begin{align} i x'(t)= c A(t)(e^{-i q(t)}) y(t) \end{align} $$ \begin{align} i y'(t)= c A(t)(e^{i q(t)}) x(t) \end{align} $$ when A(t) and q(t) are functions as follows:

\begin{align} A(t)=a cos(wt-r),\\ q(t)=b+dt+k sin(wt+r) \end{align}

where all of \begin{align} a,b,c,d,k,w,r \end{align} are constants .

and the boundary condition is as follows: $$ X(t=0)=1,\\ Y(t=0)=0 \\ $$ It might be possible to slightly simplify the solutions by considering special cases. For example assume that when one of the parameters of the theory, say $d$ has some special values say $d_0=0$ or $1$ or something. Then if in this case an analytical solution $x_0(t)$ and $y_0(t)$ can be found, in the next step you may assume that the solutions $x(t)=x_0(t)\cdot X(t)$ and $y(t)=y_0(t)\cdot Y(t)$. Hopefully the equations for $X(t)$ and $Y(t)$ can be much simpler??

Answer 1

I couldn't get a closed-form solution in general, but with $d=0$ and $r=0$ Maple finds the following solution, where $s = \sqrt{4 a^2 c^2 + k^2 w^2}$:

$$ \eqalign{ x(t) &= \frac{s+kw}{2s} e^{-i (kw - s) \sin(w t)/(2w)} + \frac{s-kw}{2s} e^{-i(kw+s) \sin(wt)/(2w)} \cr y(t) &=\frac{ac}{s} e^{ib} \left(e^{i(k w - s) \sin(w t)/(2w)} - e^{i(k w + s) \sin(w t)/(2w)}\right)} $$