# 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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