General Contents
Detailed Contents
Index
The Bernoulli Equation: Example 1
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Solve the differential equation
.
General Contents
Detailed Contents
Index
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 P(x)?
What is Q(x)?
What is the value of n?
n
= 2
What is the value of (1-
n
)
1 –
n
= 1 – 2 = -1
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
Rewrite this as the natural log of a power of x.
Evaluate
We get
Simplify this.
Use this to set up the solution of the original equation.
becomes
Do the
multiplication under the integral sign
on the right-hand side.
Do the integration.
Multiply both sides by x.
Take the reciprocal to get an expression for
y
.
Multiply numerator and denominator by 2 to clear the compound fraction.
How can we check this result?
We can differentiate to find
y
’ and substitute in our original equation.
Differentiate our result for y.
, or
, or
Substitute in the left-hand side of the original equation.
Combine the fractions.
Using the least common denominator, we get
Now we need the right-hand side of the original equation for comparison. Substitute our expression for y in that side of the original equation.
We get
Does it check?
Yes, we get the same result on each side of the original equation.
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