# 6.2b Notes

```Notes – 6.2b
Differential Equations: Growth and Decay
Assignment: Assn #3: WS 3
Learning Target: Students will be able to write the general solution and find the particular solution of a differential equation.
Students will be able to determine if an equation is a solution to a given differential equation. Students will be able to write a
differential equation that is represented by a variation statement.
Growth and Decay Models
In many instances, the rate of change of a variable y is proportional to the value of y. If y is a function of
time t, then the proportionality can be written:
Exponential Growth and Decay Model
If y is a differentiable function of t such that y &gt; 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 &gt; 0 and
exponential decay occurs when k &lt; 0.
Example 1: Finding Half-Life
a. Find the half-life of a radioactive substance with decay equation y=y0e-kt and show that the half-life
depends only on k.
b. If the radioactive decay of an element can be modeled by the differential equation
the half-life.
dy
 .0077 y . Find
dt
Example 2: Using an exponential growth model
Bacteria grow in a certain culture at a rate proportional to the amount present. Suppose there are 100
bacteria present initially and the amount doubles in 1 hour. How many will there be in an additional 0.5
hours?
 Sketch a graph
 Find k
 Find the indicated amount
Example 3: Using an exponential growth model
Sugar dissolves in water at a rate proportional to the amount still undissolved. Suppose there are 50 grams of
sugar initially and after 3 minutes only 20 grams are left. What is the total time it takes for 90 percent of the
sugar to dissolve?
*
Sketch a graph
*
Find k
*
Find the indicated amount
Example 4: Newton’s Law of Cooling
A hard-boiled egg at 98&deg;C is put in a pan under running 18&deg;C water to cool. After 5 min., the egg’s
temperature is found to be 38&deg;C. How much longer will it take the egg to reach 20&deg;C?
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