where is a constant.

It's not Bernoulli, since it's not of the form

So what is it? And how do I solve it?

Printable View

- October 31st 2010, 07:48 PMscorpion007What sort of DE is this?
where is a constant.

It's not Bernoulli, since it's not of the form

So what is it? And how do I solve it? - October 31st 2010, 08:49 PMscorpion007
Oh, perhaps it's separable?

So then I get

But how do I integrate the LHS? Partial fraction expansion? Anything easier? - October 31st 2010, 10:36 PMbandedkrait
The Differential equation is a special equation called the Riccati equation. It reduces to the Bernoulli's equation for

The solution of this equation is not very simple. One method is to make the substitution

where u(t) is a particular solution that is already known,

y is a new dependent variable.

This substitution reduces the equation to a Bernoulli's equation in y and t, and you can obtain y as a function of t. Returning to the original dependent variable gives you the actual solution as

In this case, you can take the particular solution to be one of the roots of the equation wherever is real. - October 31st 2010, 10:42 PMmr fantastic
The solution will depend on since might have:

Case 1: two distinct linear factors,

Case 2: one repeated linear factor, or

Case 3. no real linear factor

depending on the value of . For case 1, use partial fractions. For case 2, it's a standard form. For case 3 you need to complete the square and then recognise a standard form that will give you arctan.

The work hack work is left for you. (You might find it easier to start with concrete values for for each case. Note that there is only one value for case 2). - November 1st 2010, 12:29 AMscorpion007
Ah ok, so I think I have standard form, since I have lambda = 1/4 (I forgot to mention that part).

So I have

Hmm.. How do I evaluate that integral? - November 1st 2010, 01:39 AMmr fantastic
- November 1st 2010, 02:24 AMscorpion007
Oh, of course!