420
7.4. SEPARABLE DIFFERENTIAL EQUATIONS
As we will see, this will not be helpful since the left-hand side is not a product of a function
of y with
dy
dt .
Preview Activity 7.4. In this preview activity, we explore whether certain differential
equations are separable or not, and then revisit some key ideas from earlier work in
integral calculus.
(a) Which of the following differential equations are separable? If the equation is
separable, write the equation in the revised form g(y)
dy
dt = h(t).
1.
dy
dt
= −3y.
2.
dy
dt
= t y − y.
3.
dy
dt
= t + 1.
4.
dy
dt
= t
2 − y
2 .
(b) Explain why any autonomous differential equation is guaranteed to be separable.
(c) Why do we include the term “+C” in the expression
x dx =
x 2
2
+ C?
(d) Suppose we know that a certain function f satisfies the equation
f
′ (x) dx =
x dx.
What can you conclude about f ?
⊲⊳
Solving separable differential equations
Before we discuss a general approach to solving a separable differential equation, it is
instructive to consider an example.
Example 7.1. Find all functions y that are solutions to the differential equation
dy
dt
=
t
y 2 .
7.4. SEPARABLE DIFFERENTIAL EQUATIONS
As we will see, this will not be helpful since the left-hand side is not a product of a function
of y with
dy
dt .
Preview Activity 7.4. In this preview activity, we explore whether certain differential
equations are separable or not, and then revisit some key ideas from earlier work in
integral calculus.
(a) Which of the following differential equations are separable? If the equation is
separable, write the equation in the revised form g(y)
dy
dt = h(t).
1.
dy
dt
= −3y.
2.
dy
dt
= t y − y.
3.
dy
dt
= t + 1.
4.
dy
dt
= t
2 − y
2 .
(b) Explain why any autonomous differential equation is guaranteed to be separable.
(c) Why do we include the term “+C” in the expression
x dx =
x 2
2
+ C?
(d) Suppose we know that a certain function f satisfies the equation
f
′ (x) dx =
x dx.
What can you conclude about f ?
⊲⊳
Solving separable differential equations
Before we discuss a general approach to solving a separable differential equation, it is
instructive to consider an example.
Example 7.1. Find all functions y that are solutions to the differential equation
dy
dt
=
t
y 2 .