Approaching analytical solutions to a system of non-linear differential equations with sine function

Question

Developing a mathematical model for drug metabolism has led me to 3 differential equations:

1. $$ \frac {dI(t)}{dt}=-(k_f+k_a) \cdot I(t) + k_1 \cdot sin(t) \cdot G(t)$$
2. $$ \frac {dG(t)}{dt}=-k_1 \cdot sin(t) \cdot G(t)-k_2 \cdot G(t)+k_b \cdot B(t)$$
3. $$ \frac {dB(t)}{dt}=-(k_b+k_e) \cdot B(t)+k_2 \cdot G(t)+k_a \cdot I(t)$$

All k's are different constants.

Attempt:

Eq. (1) and (2) are non-linear due to the $\sin(t)$ function. I've noticed that the sum of (1) and (2) remove the non-linear portion.

However, I've tried solving the summation and am still unable to solve the DEs. I'm able to output numerical solutions via MATLAB, but am looking to solving this analytically and for advice or creative methods if this system is solvable analytically. I see that eq (3) is linear, but am unsure if that is useful.

Thanks!

Answer 1

This system of odes is obviously linear and has no generally known exact solutions. It can be rewritten in a more compact form as $$ {\dot A}(t)=WA(t)+L(t)A(t) $$ where $A=\begin{bmatrix}I(t)\ G(t)\ B(t)\end{bmatrix}$, $$ W=\begin{bmatrix}-(k_f+k_a) & 0 & 0 \\ 0 & -k_2 & k_b \\ k_a & k_2 & -(k_b+k_e)\end{bmatrix} $$ and $L=\begin{bmatrix}0 & k_1\sin(t) & 0 \\ 0 & -k_1\sin(t) & 0 \\ 0 & 0 & 0\end{bmatrix}$. Now, let us introduce the new vector $C(t)$ as $A(t)=e^{Wt}C(t)$. Then, $$ {\dot C}(t)=e^{-Wt}L(t)e^{Wt}C(t). $$ Now, the product with exponentials of matrices can be better managed by diagonalizing $W$. In any case, this is a well-known system of odes that can be generally treated by perturbation theory. The reason is the presence of the sine term.