In this post we look at examples of how differential equations can be applied to solve some real world problems.
1) A 100 gallon tank is full of pure water. Let pure water run into the tank at the rate of 2 gals/ min. and a brine solution containing 1/2 lb. of salt run in at the rate of 2 gals/min. The mixture flows out of the tank through an outlet tube at the rate of 4 gals/min. Assuming perfect mixing, what is the amount of salt in the tank after t minutes?
Let s be the amount of salt in the tank in pounds at time t. Then:
s/ 100 = concentration of salt (i.e. as a proportion of total gallons of pure water in tank initially)
Then: ds/ dt = net rate of change = (rate of gain in lbs/min - rate of loss in lbs/min)
We can further write:
ds/dt = 1 - 4s/ 100 = 1 - s/25
Writing the basic differential equation to solve: ds/ (25 - s) =
This requires integrating both sides:
Which is the basic equation for a simple harmonic oscillator, i,.e.:
w = 2 pf so:
The total force acting is therefore:
F = 100.2 - 40.2 - 2v = 60 - 2v
WHY is this? We have two negative contributions (40.2 and 2v) on the LHS because the force of friction and drag both act opposite to the direction of motion. '2v' because the drag is stated as 'twice the velocity'. The starting DE becomes:
(1000)/ g (dv/dt) = 60- 2v
Re-arranging to separate variables:
dv / (30 - v) = (32.17) dt/ 500 = 0.06434 dt
For which the solution is obtained by integration, i.e.
ln (30 - v) - ln 30 = -0.06434 t
Taking natural logs of each side:
(30 - v)/ 30 = e –0.06434t
v = 30 (1 - e –0.06434t )
After 10 seconds: v = 30 (1 - e –0.6434 ) = 30 (1 -0.5255) = 30 (0.4745)
v(10) = 14.24 feet/sec