MA354 Worksheet: Introduction to Differential Equations
Using Mathematica
Exponential Growth and Decay
dP
 kP models exponential growth when k>0 and exponential
The differential equation
dt
decay when k<0.
Exercise 1: Solve the differential equation
dP
 kP . Use separation of variables to
dt
show that P(t )  P0 e k (t t0 ) .
Exercise 2: Verify that P(t )  P0 e k (t t0 ) is a solution to
dP
 kP by taking the derivative
dt
of P (t ) by hand.
Exercise 3: Solve the differential equation
dP
 kP using Mathematica:
dt
DSolve[{p'[t]k p[t],p[0]P0},p[t],t]
Mathematica’s output:
Exercise 4: Verify that P(t )  P0 e k (t t0 ) is a solution to
using Mathematica:
D[P0*E^(k*(t-to)),t]
Mathematica’s output:
dP
 kP by taking the derivative
dt
Exercise 5: Consider P(t)= P0 e k ( t t0 ) when k = -0.2, P0 =100 and t0 = 0. The average rate
of change of P over the interval [t, t+t] is given by:
rate of change =
P(t  t ) - P(t)
.
t
(a) Using the equation above, find the rate of change when t = 10 for the following
values of t :
t = 1.0
t = 0.1
t = 0.01
t = 0.001
(b) Find the instantaneous rate of change when t = 10 using
dP
= k P.
dt
Part B: Using Octave
Now, we will model exponential decay as a stochastic process.
Exercise 1: Download and install Octave from the following web address:
http://octave.sourceforge.net/
Exercise 2: As a class, write a subroutine for the decay of 100 particles when k = -0.2,