Setting up a system for ODE solver

I have the following two equations:

dN/dt = k6*N
dE/dt = (k2*N)-(k3*N^2)-(k4*E*t)

All the k values are constants. I'm trying to solve to find what value of initial N will produce the highest value of E but I have a small issue. I'm not sure how to set up the system because the N^2 and the t already in the second equation throw me off. Can anyone help? Also how would i find the value of E that produces what I need?

Answer 1

[Walter Roberson]

You cannot find the highest value of E without knowing a boundary condition for E. Or rather you can find it symbolically in the boundary condition, but it is not a "clean" formula. If you do not have specific values for the constants then it becomes fairly difficult to understand which of the extrema leads to the maximum E.

I can show that the maximum E occurs when E(0) is maximum: one can solve and isolate out an additive constant term of exp(-(1/2)*k4*t^2)*E(0)

If we choose E(0) = 0 to remove the additive term, then by choosing the signs of k3 and k4 carefully, I can drive the maximum E value to +infinity

The extrema get pretty messy to express if k4 is positive.

[Donald]

I should probably have been more specific. I have values for all k and they are all positive. Also E(0) = 0