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The Bernoulli Equation: Example 6
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Solve the differential equation
.
What type of differential equation is this?
It is a Bernoulli equation.
What is the general form of a Bernoulli equation?
Its general form is
Must
n
be an integer?
No
What is the form of the solution?
The general solution is
For our problem, what is
?
What is
What is the value of
n
?
What is the value of
How do we start on finding the solution?
As with Linear Differential Equations, we start with finding the exponent. Here that is
Set up this integral.
Do the integration.
We get
Evaluate
We get
Use this to set up the solution of the original equation.
Equation 1
becomes
Do the integration.
We get
Equation 2
To solve for
y
, first multiply both sides by
How do we solve for
y
?
Take the
power on both sides of the equation.
Do that.
How can we check this result?
We can differentiate to compare with the original differential equation.
A simple way is to solve Equation 2 for C and differentiate implicitly. First solve for
C
.
We get
Differentiate implicitly.
How can we simplify this?
We can multiply everywhere by
to get
by itself.
Do that.
We get
, which checks.
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General Contents
Detailed Contents
Index
.