The More, The Mightier (Part 2)

Calculus Level 3

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:

\(\dfrac{dA}{dt} = -\beta B \hspace{1cm} \) and \(\hspace{1cm} \dfrac{dB}{dt} = -\alpha A \).

Constants \(\alpha\) and \(\beta\) represent the relative fighting proficiencies of "Force A" and "Force B", respectively.

Suppose that \(A=1\) and \(B=2\) at time \((t=0)\). Suppose also that \((\beta = 1)\).

Determine the value of \(\alpha\) such that the two sides fight each other for eternity, with neither side's troop strength ever being entirely reduced to zero.

Details and Assumptions: Assume that \(A\) and \(B\) can vary continuously.


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