Chapter 8: Differential Equations
Section 8.1: Solving Differential Equations
Definition: A differential equation is an equation that involves the derivative of a function.
In particular, a first-order differential equation involves the first derivative of a function. A
function that satisfies a differential equation is called a solution of the differential equation.
Definition: A separable differential equation is an equation of the form
dy
= f (x)g(y).
dx
To solve a separable differential equation, separate the variables and integrate.
dy
= f (x)g(y)
dx
dy
= f (x) dx
g(y)
Z
Z
dy
=
f (x) dx.
g(y)
A separable differential equation of the form
dy
= f (x)
dx
is called a pure-time differential equation since it depends only on the independent
variable which often represents time.
Example: Solve each pure-time differential equation.
(a)
dy
= x + cos x, where y0 = 2 for x0 = 0
dx
1
(b)
dy
= e−3x , where y0 = 10 for x0 = 0
dx
(c)
dy
1
= 2
, where y0 = 1 for x0 = 0
dx
x +1
2
Example: Suppose that the volume of a cell at time t, denoted by V (t), changes according
to the differential equation
dV
= 1 + cos t,
dt
where V (0) = 5. Find V (t).
Example: Suppose that the amount of phosphorus in a lake at time t, denoted by P (t),
satisfies the differential equation
dP
= 3t + 1,
dt
where P (0) = 0. Find the amount of phosphorus at time t = 10.
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Definition: A separable differential equation of the form
dy
= g(y)
dx
is called an autonomous differential equation.
Example: Solve each autonomous differential equation.
(a)
dy
= 2(1 − y), where y0 = 2 for x0 = 0
dx
(b)
dx
= 1 − 3x, where x(−1) = −2
dt
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Example: (Exponential Growth) Let N (t) denote the size of a population at time t ≥ 0.
Suppose that N (t) satisfies the differential equation
dN
= rN,
dt
where r is a positive constant. Solve this differential equation and determine the limiting
behavior of N (t).
Example: (Restricted Growth) Let L(t) denote the length of a fish at time t ≥ 0, and assume
the fish grows according to the von Bertalanffy equation
dL
= k(34 − L),
dt
where k is a positive constant and L(0) = 2. Solve this differential equation and determine
the limiting behavior of L(t). Determine the value of k if L(4) = 10.
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Example: Solve each differential equation.
(a)
dy
y
=
, where y0 = 1 for x0 = 0
dx
x+1
(b)
dx
x+1
=
, where x(2) = 5
dt
t−1
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Example: Use the method of partial fractions to solve
dy
= (y − 1)(y − 2),
dx
where y0 = 0 for x0 = 0.
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Example: (Logistic Growth) Let N (t) denote the size of a population at time t ≥ 0 and
suppose that N (t) satisfies the differential equation
dN
N
= rN 1 −
,
dt
K
where r and K are positive constants. Solve this differential equation and determine the
limiting behavior of N (t).
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