Applications of Differential Equations

I.  Homework

II.  Mixing Problems

Example:  Water is pouring out of a lake at the rate of 10 gallons per second.  Water from an contaminated MTBE source of 3 mg per gallon is pouring into the lake at 12 gallons per second.  The lake initially is uncontaminated and contains 1,000,000 of water.  How long will it take until the lake has 1 mg per gallon of MTBE contamination?  Hint:  Rate = Rate In - Rate Out.

III.  Hybrid Selection

Example:  Suppose that a disease resistant squirrel is born in the wild.  After t generations, the proportion of these squirrels y in the region is modeled by the differential equation

dy/dt = k(1 - y)( 2 - y)

Last year there were 15% of these disease resistant squirrels and after 4 generations there were 25%.  What percent will be disease resistant in the 10th generation?

IV.  Rate Equations

For a reaction

A + B -> C

if y is the unconverted amount of B after t seconds, then we can write

dy/dt = ky²

If initially there was 20 grams of unconverted B, and after 3 seconds, there was 15 unconverted grams of the substance, how much will be left after 4 seconds?