Diff. Eq
A 5 gallon vat is full of pure water. We start to add salt at time t = 0 to the vat at a rate of 1/2 a pound per minute. Water drains out of the vat
through a spigot at a rate of 1/2 a gallon per minute. Another pipe adds water to the vat so that the level is always at 5 gallons. Assume that the salt is evenly mixed throughout the vat at all times. 1)Set up the initial value problem for dS/dt in the above situation and 2)Find any equilibrium solutions.

Question

Diff. Eq
A 5 gallon vat is full of pure water. We start to add salt at time t = 0 to the vat at a rate of 1/2 a pound per minute. Water drains out of the vat
through a spigot at a rate of 1/2 a gallon per minute. Another pipe adds water to the vat so that the level is always at 5 gallons. Assume that the salt is evenly mixed throughout the vat at all times. 1)Set up the initial value problem for dS/dt in the above situation and 2)Find any equilibrium solutions.

johnweldon1993 · Answer

Well the volume at all times is 5 gallons v(t) = 5 gal 
The salt to begin with is 0 so s(0) = 0 

Salt is going in at a rate of 1/2 lb/min and going out at a rate of 1/2 gal/min 

Rate in - rate out 

there is 1/2 a pound of salt going in every minute so rate in = 1/2 lb/min 
Rate out...there is some amount of salt going out at the rate of 1/2 gal/min 

1/2 lb/min -  s(t)      * (1/2 gal/min) 
                    ----
                  5 gallons

So would this be 

dS + 1/2s(t) 
---   ------       = 1/2
dt         5
And continue on finding the integration factor? Am I on the right track here?