Developing a mathematical model for drug metabolism has led me to 3 differential equations:
- $$ \frac {dI(t)}{dt}=-(k_f+k_a) \cdot I(t) + k_1 \cdot sin(t) \cdot G(t)$$
- $$ \frac {dG(t)}{dt}=-k_1 \cdot sin(t) \cdot G(t)-k_2 \cdot G(t)+k_b \cdot B(t)$$
- $$ \frac {dB(t)}{dt}=-(k_b+k_e) \cdot B(t)+k_2 \cdot G(t)+k_a \cdot I(t)$$
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!