The Bernoulli Equation: Example 2

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?

What is the value of (1-n)

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.

becomes
Equation 1

Set up the integral on the right-hand side for integration by parts.

We’ll take care of the multiplying fraction later:

What should we choose for u?

If we choose u = x, the problem will get simpler.

List the parts.

Substitute these parts.

Do the integral on the right-hand side and simplify.

We get

Substitute these results into Equation 1 above.

Multiply both sides by

Equation 2

Take the power to get an expression for y.

How can we check this result?

The simplest way is to differentiate implicitly our Equation 2
First differentiate the left-hand side implicitly.

Using the Power Rule, we get

Now differentiate the right-hand side of our equation for

Set these results equal to each other.

Solve for the expression

Equation 3

Notice that the original equation can be rewritten to contain this same expression.
Divide both sides of
by

We get

Simplify by using

We get

We can now use the results from Equation 3 and  Equation 2 in the left-hand side of Equation 4 to see if our result is correct.

We get

Two pairs of terms eliminate leaving –[-x] = x, which checks with the right side of Equation 4.

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