General Contents
Detailed Contents
Index
Ordinary Differential Equations: Separation of Variables: Example 3
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Solve
by separation of variables.
General Contents
Detailed Contents
Index
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 multiply both sides by
.
Do it.
We get
as our
separated differential equation
.
Are the variables separated?
Yes.
What will the form of the integral on the left side be?
It will be an exponential.
Set up that integral using
w
as a temporary variable for the exponent.
We get
,
, and the integral becomes
We’ll take care of the constant of integration when we integrate the right side of the separated differential equation.
Do the integral.
We get
Re-express this in terms of
y
.
We get
Integrate the right-hand side of our separated differential equation.
We get
Combine these results to get the solution of the differential equation.
We get
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by separation of variables.