MAT 265: EXPONENTIAL GROWTH/DECAY
DIFFERENTIAL EQUATIONS
J. BEAUMONT, A.CAINE
1. Exponential Growth/Decay
Possibly the most common, albeit simplified, model for natural phenomena is what’s called Exponential Growth.
Definition 1. Let y(t) be a differentiable function. If there exists a k∈Rsuch that
y y
′(t) = k·(t)
is said to demonstrate .
In layman’s terms this says, “The rate at which ygrows is proportial to how much of ythere is.”
Example (Finance: Compound interest).If you are ridiculously lucky, you can find an investment/savings ac-
count, that “compounds continuously” 3% interest. This means the bank pays you money to leave your money
with them. But how much will you be paid? The amount you’re paid is a percentage of how much you have
invested, i.e.
the rate at which your money grows is proportial to how much money you invest. In this case that percentage
is
Exercise 1. In the example above. Let ) represent the number of dollars you have invested at timeX(t t. The
bank pays you by adding money to your investment at 3% interest compounded continuously. What do you think
kshould be so that
X X
′(t) = k·(t)?
Example (Chemistry: Radioactive Decay).Atoms don’t like to be radioactive (they don’t like to have an imbal-
anced number of protons, neutrons, and/or electrons) and over time the imbalance corrects itself. For example,
carbon prefers an atomic mass of 12; with 6 protons and 6 neutrons, but occurs naturally when a carbon atomC14
has 8 neutrons. Since the atom doesn’t like this state, it will emit a beta particle (an electron) and becomeC14
an N14 atom. Notice, the total mass stays the same (14), the atom is just no longer radioactive.
The half life of is about 5730 years. This means that if a dionsaur fossil contains ) kilograms of carbonC14 C(t
14, roughly 0.01209681% of it will decay (i.e. convert to Nitrogen 14) every year.
Exercise 2. Write the exponential growth equation for C(t) with the stats given above. [Hint: Careful with the
sign of ]k
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MAT 265: EXPONENTIAL GROWTH/DECAY DIFFERENTIAL EQUATIONS 2
Exercise 3 (Biology: Population Growth).Suppose there is a strain of bacteria that reproduces at a rate of 20%
per minute. This means, if you start with P(t) number of bacteria at time , we expect there to be 20% more onet
minute later.
Write the exponential growth equation for P( )t
Up until now we have been making Differential Equations to describe how a variable changes. That is, we have
been writing an equation that relates derivative of a function to the function itself. In this case it’s the exponential
growth equation y′=k·y. However, it would be much more useful to have a formula for the variable of interest.
That is, instead of X′( 03t) = . X(t), I want to know a formula for X(t) =the amount of money I have in the
account at time .t
To do this we have to solve the differential equation.
Solving The Exponential Growth Equation. If I claim a solution to x2+ 7x+ 1 = 19 is x= 2, how do you
know it’s right? Don’t over think this, essetially, I’m asking how you check your work.
So, if our answer satisfies the equation, we call it a solution. This is the same idea to solving a differential
equation.
Exercise 4. Let f(t) = a·et, where is constant.a
(a) Compute ) (you must use the limit definition of derivative here, until we learn the “shortcut”).
f′(t
(b) From your answer above, try to make an educated guess on what ) should be, if
g′(t g(t) = a·ekt , where a
and kare constant (don’t forget the chain rule!)
MAT 265: EXPONENTIAL GROWTH/DECAY DIFFERENTIAL EQUATIONS 3
Exercise 5. Let f(t) = ek·t
(a) Compute ).
f′(t
(b) How does f( ( )?t) relate to f′t
(c) Find ) so that
X(t X′(t) = .03 ·X( ).t
(d) Find ) so that
C(t C′( ).t) = .0001209681 ·C(t
Remark. Later this semester you’ll learn a technique called integration, which in some since is the inverse of
differentiation, and will allow you to compute the solution to such differential equation.
In all of f(t), X( ) (t , C t), we did not worry about an initial condition. That is, we did not specify an initial
value, f(0) (0), X , C(0). Technically, this means the previous exercise was not well posed and indeed has more
than one solution. This is easily remedied.
Example. Let X(t) = 27000e.03t
(a) Compute X′( ).t
(b) Does satisfy the exponential growth formula?X
(c) What is the value of (0)?X
Exercise 6. If y(t) = Aekt, where A, k are constants, what is (0)? Solve fory A when .t=b
If a differential equation specifies an it is called an initial value problem. The next two exercisesinitial condition
may be referred to as differential equations, initial value problems, or exponential growth problems. All are correct.
Exercise 7. (a) If at time zero there are 10 bacteria on my foot and every second the poputation grows by 1.23%,
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find an equation to model the number of bacteria as a function of time (please specify units).
(b) Solve y′( 1]t) = iπ ·y(t), y(−1) = 2 [Hint: e−iπ =−
