Lecture Notes in Differential Equations - Bruce E. Shapiro

45
Exponential Runaway
First we consider τ > 0. For convenience we will choose t 0 = 0. Then
y = C + (y 0 − C)e t/τ (6.14)
The solution will either increase exponentially to ∞ or decrease to −∞
depending on the sign of y 0 − C.
Example 6.1. Compound interest. If you deposit an amount y 0 in the
bank and it accrues interest at a rate r, where r is measured in fraction per
year (i.e., r = 0.03 means 3% per annum, and it has units of 1/year), and t
is measured in years, then the rate at which the current amount on deposit
increases is given by
dy
= ry (6.15)
dt
Then we have C = 0 and τ = 1/r, so
y = y 0 e rt (6.16)
So the value increases without bound over time (we don’t have a case where
y 0 < because that would mean you owe money).
Suppose we also have a fixed salary S per year, and deposit that entirely
into our account. Then instead of (6.15), we have
dy
dt
= ry + S = r(y + S/r) (6.17)
In this case we see that C = −S/r, the negative ratio of the fixed and
interest-based additions to the account. The solution is then
y = − S (
r + y 0 + S )
e rt (6.18)
r
We still increase exponentially without bounds.
Now consider what happens if instead of depositing a salary we instead
withdraw money at a fixed rate of W per year. Since W causes the total
amount to decrease, the differential equation becomes
dy
dt
= ry − W = r(y − W/r) (6.19)
Now C = W/r. If W/r < y 0 then the rate of change will be positive initially
and hence positive for all time, and we have
y = W (
r + y 0 − W )
e rt (6.20)
r