General Contents

Detailed Contents

Ordinary Differential Equations: Separation of Variables: Example2

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Solve

by separation of variables. Find the general solution.

General Contents

Detailed Contents

How can we “move” y from the first term to the second?

We can divide both terms by y.

Do it.

We get

Are the variables separated?

No. We need all of the x dependence in the first term.

How can we accomplish this?

We can multiply all terms by x.

Do it.

We get

.

Now the variables are separated. How do we simplify the second term?

Do the division and get

In order to integrate the first term, we need to substitute

What will we get for du?

, or

How will the first term change?

It becomes

Rewrite the differential equation in terms of u.

We get

Do the integration.

We get

Rewrite this result in terms of x, using the definition of u from above.

We get

We want to solve for y. It will be easier if we multiply by 2.

We can use some properties of logarithms here. Change the 2 multiplier to an exponent.

This is the general solution for y in implicit form.

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