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Solutions and Commentary

Solutions and Commentary
1. Assuming the combat effectiveness of the combatants is equal, when we add the two equations together, we get


Recall that the sum of derivatives is the derivative of a sum. So we actually have
This equation is a variation of the basic differential equation for exponential functions  , where  . The solution to     is  , and so the solution to our equation is
This means that the total number of ships in the battle decreases exponentially over the time of the battle.

2. In a similar manner, we can consider the difference in the two defining differential equations.   is equivalent to 

and so
In the course of the battle, the difference in the number of ships that remain in the winning fleet A and those remaining in the losing fleet B increases exponentially.

3. We now have two equations in two unknowns, A and B. Solve for A by adding the two equations from questions (1) and (2) together to eliminate B. This yields


and

Solving for B, we find that


4. If    and  , then  . This shows that the second derivative of A is proportional to A. What function is this? Only the exponential function has this property. However, there are two solutions, since the exponent could be either positive or negative. Both    and   satisfy the differential equation  . The general solution, then, is the sum of the two. So    is the general solution to the second-order differential equation  .

Now,  . This means that  . By integration, we find that  .

Since we have     and  , we can solve for     and  . Notice that if  , we have the same solutions as in question (3).

5. We know that     and  . So, from the chain rule and inverse function rule, we have     and  . This is a separable equation, so  , and it follows that  .

The initial conditions give us  , so  . The end of the battle occurs when  . Substituting into the equation above, we find that the number of ships in fleet A at the end of the battle is  .

If fleet B wins, then the expected number of surviving vessels is  . By using this equation, we can determine the expected number of ships remaining in the winning fleet at the end of the battle. For example, two equal forces ( ) with     and     would result in a victory for A with 13 ships remaining after the battle. This surprisingly large number is a result of the two exponential functions above. The total number is decreasing exponentially, but the difference in the size of the fleets is increasing exponentially.

6. If Nelson's 27 ships fought a conventional battle against the 34 ships in the French/Spanish Armada with  , then  , and Nelson would lose. The expected number of ships remaining in the winning French/Spanish fleet would be     or 14 ships. (We will consider a "fractional ship" as still capable of fighting if the fraction is greater than one-half.) This would not be a good day for Admiral Nelson and the British.

7.
First Battle (Fleet A, the French/Spanish Armada, Wins) Second Battle (Fleet B, the British, Wins) Final Battle Winner
British French/
Spanish
Survive British French/
Spanish
Survive British French/
Spanish
Survive  
4 17 16 23 17 18 18 16 11 British
5 17 16 22 17 16 16 16 8 British
6 17 16 21 17 15 15 16 6 British
7 17 15 20 17 14 14 15 5 British
8 17 14 19 17 12 12 14 2 French/
Spanish
9 17 13 18 17 10 10 13 6 French/
Spanish


The number in the Survive column is the number of ships remaining after the battle. For example, looking at the first row, the British have 27 ships and the Armada has 34. The British split the battle into a fight of 4 British ships against 17 Armada. They lose, with the Armada having 16 ships left. The remainder of the British fleet (23 ships) fights a battle against the remaining 17 in the Armada. The British win with 18 ships surviving. Those 18 winning British ships from the second battle fight the 16 winning Armada ships from the first battle. The British win with 11 ships remaining at the end.

The table shows that Nelson has many options. However, he will want the first battle (the one he loses) to last as long as possible, so that the second battle will have the best chance of concluding before the survivors of the first battle can rejoin the fight. Having 7 ships in the first battle and 20 in the second allows Nelson to defeat the larger fleet with an expected 5 ships surviving.

The actual battle had more ships and did not follow this outline, but Nelson's original, creative strategy changed naval history.







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