CHAPTER 3
Differential Equations
3.1. Differential Equations and Separable Equations
3.1.1. Population Growth. The growth of a population is usu-
ally modeled with an equation of the form
dP
= kP ,
dt
where P represents the number of individuals an a given time t. This
model assumes that the rate of growth of population is proportional to
the population size.
A solution to this equation is the exponential function:
P (t) = Cekt .
Check: P 0 (t) = kCekt = kP (t).
A more realistic model takes into account that any environment has
a limited carrying capacity K, so if P reaches K the population stops
growing. The model in this case is the following:
µ ¶
dP P
= kP 1 − .
dt K
This is called the logistic differential equation.
3.1.2. Motion of a Spring. Consider an object of mass m at the
end of a vertical spring. According to Hook’s law the restoring force of
a spring stretched (or compressed) a distance x from its natural length
is
F = −kx ,
where k is a positive constant (the spring constant) and the negative
sign expresses that the sense of the force is opposite to the sense of
the stretching. By Newton’s Second Law (force equals mass times
74
, 3.1. DIFFERENTIAL EQUATIONS AND SEPARABLE EQUATIONS 75
acceleration):
d2 x
m = −kx ,
dt2
or equivalently:
d2 x k
= − x.
dt2 m
This is an example of a second order differential equation because
it involves second order derivatives.
3.1.3. General Differential Equations. A differential equation
is an equation that contains one or more unknown functions and one
or more of its derivatives. The order of the differential equation is the
order of the highest derivative that occurs in the equation.
3.1.4. First-order Differential Equations. A first-order differ-
ential equation is an equation of the form
dy
= F (x, y) ,
dx
where F (x, y) is a function of x and y. A solution of the differential
equation is a function y(x) such that y 0 (x) = F (x, y(x)) for all x in
some appropriate interval.
Example: Consider the following differential equation:
dy 2y
= .
dx x
A possible solution for that equation is, for instance, y = x2 , because
dy
= y 0 (x) = (x2 )0 = 2x ,
dx
and
y x2
2 =2 = 2x ,
x x
2 y(x)
hence y 0 (x) = for all x 6= 0.
x
Differential Equations
3.1. Differential Equations and Separable Equations
3.1.1. Population Growth. The growth of a population is usu-
ally modeled with an equation of the form
dP
= kP ,
dt
where P represents the number of individuals an a given time t. This
model assumes that the rate of growth of population is proportional to
the population size.
A solution to this equation is the exponential function:
P (t) = Cekt .
Check: P 0 (t) = kCekt = kP (t).
A more realistic model takes into account that any environment has
a limited carrying capacity K, so if P reaches K the population stops
growing. The model in this case is the following:
µ ¶
dP P
= kP 1 − .
dt K
This is called the logistic differential equation.
3.1.2. Motion of a Spring. Consider an object of mass m at the
end of a vertical spring. According to Hook’s law the restoring force of
a spring stretched (or compressed) a distance x from its natural length
is
F = −kx ,
where k is a positive constant (the spring constant) and the negative
sign expresses that the sense of the force is opposite to the sense of
the stretching. By Newton’s Second Law (force equals mass times
74
, 3.1. DIFFERENTIAL EQUATIONS AND SEPARABLE EQUATIONS 75
acceleration):
d2 x
m = −kx ,
dt2
or equivalently:
d2 x k
= − x.
dt2 m
This is an example of a second order differential equation because
it involves second order derivatives.
3.1.3. General Differential Equations. A differential equation
is an equation that contains one or more unknown functions and one
or more of its derivatives. The order of the differential equation is the
order of the highest derivative that occurs in the equation.
3.1.4. First-order Differential Equations. A first-order differ-
ential equation is an equation of the form
dy
= F (x, y) ,
dx
where F (x, y) is a function of x and y. A solution of the differential
equation is a function y(x) such that y 0 (x) = F (x, y(x)) for all x in
some appropriate interval.
Example: Consider the following differential equation:
dy 2y
= .
dx x
A possible solution for that equation is, for instance, y = x2 , because
dy
= y 0 (x) = (x2 )0 = 2x ,
dx
and
y x2
2 =2 = 2x ,
x x
2 y(x)
hence y 0 (x) = for all x 6= 0.
x
