1. ## Differential Equations.....forming them!!

Hi,

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?

Thanks!! xx

2. Originally Posted by emma_louise87
Hi,

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?

Thanks!! xx
Presumably you learned in Calculus that the "partial derivative" of a function with respect to a given variable is equal to the rate of change of the function while other functions are being held constant. That is exactly what is going on.

"An increase of number of escort vessels from 6 to 9 led to a reducton of losses by about 25 per cent."
$\frac{\partial L}{\partial E}= \frac{-.25}{3}L$
where L is the number of Losses and E is the number of escort vessels.

"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 %."
$\frac{\partial L}{\partial S}= \frac{-.56}{54-32}L$
where L is again the number of Losses and S is the size of the convoy.

3. Thank you! I was thinking along those lines but it has been aaaaages since i've done anything like it so i wasn't sure if i was right or not!

4. Ok.....right i now have another question for you....i think i'm being a bit dense today. But how would you compare the derivatives?

Blackett goes on to compare them and gets solutions such as

" the number of escorts would have to be increased from 6 to 11 to reduce the percentage loss of ships in small convoys of average size 32 to the losses in larger convoys of average size 54."

Now again i may be being dense but using the information given one can assume a linear realtionship and get that L decreases by 25% for every 3 additional escorts. This however does not give the answer that 11 escorts will reduce the losses by 56%.

Is there a way of comparing derivatives that i appear to be unaware of?