Solving the Interesting Problem of the Last Bit of Water in a Bottle

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SUMMARY

The discussion centers around solving a differential equation related to the volume of backwash in a water bottle after multiple sips. The equation is defined as dy/dx = 0.05 - 20(y / (1000 - (20 - 0.05)x)), where y represents the backwash volume and x the number of sips taken. The problem is framed within a 1000mL bottle, with each sip being 20mL and contributing 0.05mL of backwash. A suggested change of variables, x' = 1000 - (20 - 0.05)x, is proposed to facilitate solving the equation, referencing resources on exact differential equations for further assistance.

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There is a popular myth that after drinking a bottle of water, the last bit is mostly backwash. Well I decided to try and test it, but got stumped.

Lets call y the amount of backwash in the bottle
Lets call x the number of sips taken
The volume of the bottle will be 1000mL
Assume each sip is 20mL
Assume that each sip backwashes 0.05mL into the bottle

dy/dx= 0.05 -20( y / (1000 -(20-0.05) x ) )

I can't separate variables here, so I do not know what to do. This is not a homework problem, I was just wondering if anyone could help me solve this differential equation. At 51 sips there will be nothing left in the bottle.
 
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I tinkered with it a bit... try making the change of variables [itex]x' = 1000 - (20 - 0.05)x[/itex], then look at http://en.wikibooks.org/wiki/Differential_Equations/Exact_1 . I didn't take the calculation all the way through but it looks solvable that way.
 
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