# Solving a System of Differential Equations

I am attempting to solve and graph the solution to an initial value problem containing a system of differential equations. If I am remembering calculus correctly, its properties (nonlinear, ordinary, no explicit appearance of the independent variable time) classify it as a 'time-invariant autonomous system'.

I want to model a situation in which a microorganism in a bioreactor at some concentration 'X' is growing exponentially 'dX/dt = u*X' by consuming a substrate at some concentration 'S'.

Substrate consumption '-dS/dt' and microorganism growth rate 'dX/dt' are related by a constant representing the yield of biomass on substrate, Y. If I stop here and leave the specific growth rate 'u' a constant, I can evaluate this just fine with **desolve_system()**:

The problems start when I make 'u' (specific growth rate) a function of 'S' (substrate concentration):

u = u_max*S/(K_m+S).

I make the assumption that this must be evaluated numerically, and switch to **desolve_system_rk4()**. I get an "Error executing code in Maxima", which also states "in definition f_rk_4, found bad argument 'X(t)"

Where am I going wrong? Can I use desolve_system_rk4() to evaluate this system, and I'm just making a syntax error?

Thank you for your patience - I just started using SAGE yesterday.

- Brandon