General Contents
Detailed Contents
Index
Ordinary Differential Equations: Separation of Variables: Example 4
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Solve Solve
by separation of variables.
General Contents
Detailed Contents
Index
Rewrite the equation in differential form.
Are the variables separated?
Yes.
What technique of integration can we use here?
Substitution.
On both sides of the equation?
Yes.
Set up the integral on the left side using
u
as a temporary variable for the quantity in parentheses.
Let
Determine
du
in terms of
du
.
Solve for
y*dy
, which is the quantity we need to replace.
Combine these results to express the integral in terms of our new variable.
Do the integration.
We get
We’ll include the constant of integration when we do the integral on the right side.
Express this result in terms of
y
.
We get
Now let’s consider the other side of the equation,
. What shall we substitute a new variable
w
for?
Let
Determine
dw
in terms of
dx
.
Rewrite the integral in terms of
w
.
Do the integral.
Express this result in terms of x.
We get
Combine all of these results to express the solution of the differential equation.
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by separation of variables.