I'm doing a history of maths project based on operational research during WWII. Basically I have found old general methods that were used by scientists such as PMS Blackett during the 1940's and i need to provide examples of how they work.
I havn't done much applied maths and am not sure how to form differential equations in this way.
Here is an example given by PMS Blackett;
"Variation of Loss of Ships with Various Parameters
The 1st step in analysis is to break down the statistics of loss in such a way as to give their variation with the main variables of immediate interest. There variables are;
Number of Escorts: An increase of number of escort vessels from 6 to 9 led to a reducton of losses by about 25 per cent.
Size of Convoy: An increase of size from an average of 32 to 54, was associated with a decrease of fractional losses (i.e. ships sunk/ships sailed) from 2.5% to 1.1%, i.e. a reduction of losses of 56 %."
......etc etc there are a few more variables given.....Blackett then goes on to say that "since in each f these derivations it was verified that the average value of the other variables was about constant the four results represent in effect four partial derivatives". and then using these derivatives they would find the optimum values for the different variables.
I am really stuck on how to form these partial derivatives....I wondered if anyone could lend a hand?