Lecture Notes in Differential Equations - Bruce E. Shapiro

46 LESSON 6. EXPONENTIAL RELAXATION
Figure 6.1: Illustration of compound interest at different rates of withdrawal
in example 6.1. See text for details.
1200
1000
800
600
400
200
5 10 15 20
If instead W/r > y 0 , the initial rate of change is negative so the amount will
always be less than y 0 and your net deposit is decreasing. The term on the
right is exponentially increasing, but we can only make withdrawals into
the balance is zero. This occurs when the right hand side of the equation
is zero.
( )
W W
r = r − y 0 e rt (6.21)
or at a time given by
t = − 1 (
r ln 1 − y )
0r
W
(6.22)
So if you withdraw money at a rate faster than ry 0 you money will rapidly
go away, while if you withdraw at a slower rate, your balance will still
increase. Figure 6.1 shows the net balance assuming a starting deposit of
$1000 and a fixed interest rate of 5% per annum, for rates of withdraw
(bottom to top) of $250, $100, $60, and $40 per year. For the example the
break-even point occurs when W = ry 0 = (.05)(1000)=$50.
Exponential Relaxation
When τ < 0 we will see that the behavior is quite different - rather then
exponential run-away, the solution is pulled to the value of C, whatever the
initial value. We will call this phenomenon exponential relaxation.
As before, it is convenient to assume that t 0 = 0 and we also define T =
−τ > 0 as a positive time constant. Then we have
y = C + (y 0 − C)e −t/T (6.23)