BSU Math 333 (Ultman)
Worksheet: Separable Differential Equations
Prerequisites
In order to learn the new skills and ideas presented in this worksheet, you must:
Use algebraic manipulation to re-write a differential equation in a particular form.
Use algebraic manipulation to solve for constants.
Determine when an equation is undefined.
Integrate functions of a single variable.
Goals
In this worksheet, you will:
Identify separable differentail equations.
Solve separable differential equations and initial value problems.
Determine the interval(s) (with respect to the independent variable) on which a solution to a
separable differential equation is defined.
Describe the behavior of a solution to a separable differential equation as the independent
variable tends toward infinity.
Warm-Up
Consider the first order differential equation:
x2
dy
=
,
dx
y(1 + x3 )
where y is a function of x. Rewrite this equation so that one side of the equation is a function only
dy
of the variable y multiplied by dx
, and the other side is a function only of the variable x. In other
dy
words, write it in the form N (y) dx = M (x).
A differential equation that can be written this way is called separable.
BSU Math 333 (Ultman)
Worksheet: Separable Differential Equations
1
Separating Variables
1. For each of the following equations, separate the variable if possible. Otherwise, explain why
the equation non-separable.
(a)
2x
dy
=
dx
(y + x2 y)2
(b)
xy + 2
dy
=
dx
x+1
(c)
dy
+ 3ty = 0
dt
(d)
dy
+ 3ty = sin t
dt
BSU Math 333 (Ultman)
Worksheet: Separable Differential Equations
2
Solution Technique: Separation of Variables
Background: From a technical point of view, the solution technique for separable differential
equations involves “undoing” the chain rule. This is the idea:
◦ Separate the variables, so the equation is in the form N (y)
◦ Find a function F (y) such that
dy
= M (x).
dx
d
dF dy
dy
F (y) =
= N (y) .
dx
dy dx
dx
◦ You can now integrate both sides with respect to x:
ˆ
ˆ
ˆ
d
dy
F (y) =
F (y) dx = N (y)
= M (x) dx.
dx
dx
So the general
solution to the differential equation is (possibly implicitly) defined by
ˆ
F (y) = M (x)dx. If possible, you can solve for y to find an explicit soluton.
In practice, there is a less technical approach one can adopt. This involves treating the derivady
tive dx
as the quotient of the differentials dy and dx (it’s not, really, but we will pretend it is),
and treating the differentials as though they are numbers that can be manipulated according
the rules of arithmetic. Using this method, the algorithm for solving a separable equation is:
Step 1: Separate the variables, so the equation is in the form:
N (y)
dy
= M (x).
dx
Step 2: Multiply both sides by dx:
N (y) dy = M (x) dx.
Step 3: Integrate both sides, with respect to y on the left, and with respect to x on the right:
ˆ
ˆ
N (y) dy = M (x) dx.
The result of the final step defines the general solution to the separable differential equation.
If possible, you can solve for y to obtain an explicit formula for the solution. Otherwise, the
solution is defined implicitly.
BSU Math 333 (Ultman)
Worksheet: Separable Differential Equations
3
2. Use the algorithm to find the general solution for the differential equation in the warm-up
question, and for any equations in #1 that are separable.
BSU Math 333 (Ultman)
Worksheet: Separable Differential Equations
4
Initial Value Problems and Intervals of Definition
Recall the separable differential equation from the warm-up:
dy
x2
=
.
dx
y(1 + x3 )
Suppose we want to find the particular solution for this differential equation, given the initial
condition y(0) = 4.
In question #2, you should have found the general solution to this equation:
r
2 ln 1 + x3 + c
y(x) = ±
3
where c is an arbitrary constant.
3. Use the initial condition to solve for c, and determine whether to use the positive or negative
root.
4. Determine the values of x on which the solution is defined, by looking at the logarithmic and
square root functions. Write these using interval notation. This is the interval(s) on which
your solution is defined.
5. Take the limit of your solution from question #3 as x → ∞. To what value does your solution
tend as x tends to infinity?
6. Finally, plot the direction field and look at the solution that passes through the point (0, 4).
Does this agree with the solution you found in question #3?