differential equations: growth & decay (6.2) march 16th, 2012
TRANSCRIPT
Differential Equations: Growth & Decay (6.2)
March 16th, 2012
I. Differential Equations
To solve a differential equation in terms of x and y, separate the variables.
Ex. 1: Solve the differential equation.
a.
b.
y '=x
3y
dy
dx=4 −y
II. Growth & Decay Models
In applications, it is often true thatdy
dt=ky
the rate of change of ywith respect
to time
is proportionalto y
Thm. 6.1: Exponential Growth & Decay Model: If y is a differentiable function of t such that y>0 and y’=ky, for some constant k, then y=Cekt.C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k>0, and exponential decay occurs when k<0.
Ex. 2: Write and solve the differential equation that models the statement, “The rate of change of N is proportional to N. When t=0, N=250 and when t=1, N=400.” What is the value of N when t=4?
*Recall that for radioactive decay applications, a half-life of 20 years will allow you to find the decay model y=Cekt with the equation 1/2=e20k.
*Also, for compound interest applications, the models for growth are when interest is compounded continuouslyand when compounded n times per year.P=principal amount, r=rate, t=time (in years)
A=Pert
A=P 1+rn
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nt