General Contents
Detailed Contents
Index
Ordinary Differential Equations: Separation of Variables: Example 5
If you find this page helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
Solve
by separation of variables.
Rewrite the equation in differential form.
How can we “move” the
y
-dependence from the right-hand side to the left-hand side?
We can factor and divide.
Do it.
We get
, or
Are the variables separated?
Yes.
What will the form of the integral on the left side be?
It will be a natural logarithm.
Set up that integral using
w
as a temporary variable for the denominator.
We get
, and
, and the integral becomes
Do the integral.
We get
We’ll take care of the constant of integration when we integrate the right side of the separated differential equation.
Re-express this in terms of
y
.
We get
Set up the integral of the right-hand side of our differential equation using
u
as a temporary variable for the denominator.
We get
, and
, and the integral becomes
What type of function do we expect on the right side?
A natural logarithm.
Do the integral.
Rewrite the result in terms of
t
.
Combine these results to express the solution of the differential equation.
Can we simplify by “undoing” the logarithms?
Yes.
How?
Replace C by
, where
is also a constant. Then we get a single log on the right side.
Do that.
Now take the inverse of the natural logs.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
General Contents
Detailed Contents
Index
by separation of variables.