Ordinary Differential Equations: I\'m having issues with the Lanchester equation

Question

Ordinary Differential Equations: I'm having issues with the Lanchester equations for ancient warfare. Question is modeled on Incas vs. Spaniards. Incas had 7,000 troops, Spaniards had a total of 168, but better weapons and armor. The Spanish won by killing all Incas in 8 hours, and lost no men. We are *not* to differentiate. The equations are dS/dt=-a*S(t)*I(t) and dI/dt=-b*S(t)*I(t). We are to solve for a and b (killing constants), and then time-step through the battle. I'm stuck on solving for a and b. My numbers feel way off. so, for a, I got a=(-dS/dt)/(S/I), were dS is 0, because it never changes, dt is 8 (end time), S is (168+168)/2 =168 (avg number of Spaniards), and I is (7000+0)/2=3,500. The 0 is throwing me for some reason. When I solve for b with the same logic, I get a honking huge number in the tens of thousands, which also doesn't feel right. Is my number sense off? Where am I going wrong? Please help.

Explanation / Answer

were dS is 0,

then

a=(-dS/dt)/(S/I)

is :0

so cheek it,

dt is 8 (end time), S is (168+168)/2 =168

and I is (7000+0)/2=3,500.

  dI/dt=-b*S(t)*I(t).=

b=(-dL/dt)/(S/I)

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