Separable D.E.’s  -  Growth and Decay Applications

 1 In a Nut Shell:  There are several mathematical models to represent growth and decay. Two of the simpler ones are given below. 2 Differential equation involving “growth”                     dy/dt   =    K y   ;  here rate of change of  y  is proportional to itself       subject to    y(0)  =  yo     which represents the initial condition of y  at  t = 0                   K  =  proportionality constant   Applications:   a.  population growth                           b.  annual rate (income)                     dy/y   =  K dt   (separate variables   y  and  t  and  integrate)                  ln y  =  K t  +  C1    ,    C1     is the constant of integration                     y  =  e Kt + C1     =   C e Kt 3 Differential equation involving “decay”  (negative growth)                    dy/dt   =   - K y   ;  here rate of change of y  is proportional to its       subject to    y(0)  =  yo     which represents the initial condition of y  at  t = 0                   K  =  proportionality constant i.e.                 K  =  decay constant                           K  =   sales decay constant                 K  =  drug elimination constant   Applications:   a.  radioactive decay  (radiocarbon dating)                           b.  annual rate (income)                     dy/y   =  - K dt   (separate variables   y  and  t  and  integrate)                  ln y  =  - K t  +  C1    ,    C1     is the constant of integration                     y  =  e -Kt + C1     =  C e –Kt