The More, The Mightier

Lanchester's Square Law can be used to roughly describe the way in which two opposing military forces change over time during battle. Suppose the number of troops in "Force A" is $A$ , and the number of troops in "Force B" is $B$ .

The rates of change in troop strength (numbers of troops) over time are given by: $\dfrac{dA}{dt} = -B$ and $\dfrac{dB}{dt} = -A$ .

Suppose that $A=2$ and $B=1$ at time $(t=0)$ . What is the value of $A$ at the moment in time at which $A=100B$ ?

Give your answer to 3 decimal places.

Details and Assumptions :

• Assume that $A$ and $B$ can vary continuously, and that they are multiples of some standard measure.

• Evidently, the larger force has a distinct advantage if all else is equal.

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For those who are interested, you can read up Lanchester's Law here .