System of nonlinear differential equations

Question

Let $p(x)=10*x^5 - 21*x^4 + 18*x^3 - 13*x^2 + 6*x$. Does the system

$$\begin{aligned}\frac{dm}{dt}&=K+M*p\left(\frac c m\right)\\\frac{dc}{dt}&=K+N*p\left(\frac c m\right)\end{aligned}$$

have an analytical solution? If so, how can it be reached?

Answer 1

Introduce the variables $\tilde{c} = c - Kt, \; \; \tilde{m} = m - Kt$. The ODE become: $$ {d\tilde{m} \over dt} = M p \left( {\tilde{c} + Kt \over \tilde{m} + Kt}\right), \quad {d\tilde{c} \over dt} = N p \left( {\tilde{c} + Kt \over \tilde{m} + Kt}\right). $$ Now take their ratio: $$ {d \tilde{m} \over d \tilde{c}} = {M \over N} = \mbox{const.} $$ This should get you started.