Partial Fractions Part II

1. Logistics Growth
   It has been said that the total population of South Lake Tahoe should never exceed 30,000 people.  If in 1980 the population was 15,000 and in 2000 it was 20,000, when will the population reach 25,000.    
   
   Solution:  We make the assumption that the rate of increase of the population is proportional to the product of the current population and 30,000 minus the current population. That is:
   dP/dt = kP(30,000 - P)
   This is a separable differential equation.  Separating gives
   dP/(P(30,000 - P) = kdt
   Now integrate both sides.  The right hand side is kt + C.  To integrate the right hand side, use partial fractions:
   1/(P(30,000 - P)) = A/P + B/(30,000 - P) Multiplying by the common denominator:
   1 = A(30,000 - P) + BP  
   P = 0:  1 = 30,000A or A = 1/30,000
   P = 30,000:  1 = 30,000B or B = 1/30,000
   Now integrate to get
   1/30,000 lnP - 1/30,000 ln(30,000 - P) = kt + C or
   ln[P/(30,000 - P)] = at + b
   When t = 0, P = 15,000, so
   b = 0
   When t = 20, P = 20,000, so
   20a = ln2 or a = (ln2)/20, hence
   ln[P/(30,000 - P)] = (t ln2)/20 Finally
   ln[5] = (t ln2)/20 or
   t = (20ln5)/(ln2) = 46.4 or the year 2026
   

2. Partial Fractions Exercises

   Exercises:

   A)  int (x - 2)/(x² +3x - 4) dx

   B)  int x/(x + 1)(x² +4) dx

   C)  int (2x + 3)/[x²(x - 3)] dx

   D)  int (6x + 1)/(3x² + x - 2) dx

   We will do more if time permits.