Nonlinear First Order Differential Equations

[JoshHolloway]

Hello. I am taking a self study diff e course, and I have run into a problem with no one to ask for help. Here is the problem:
$$y\prime=1+x+y^2+xy^2$$

The question asks to find the general solution. I simply don't understand how to solve this problem. Here is the direction I am going in:
$$dy=(1+x+y^2+xy^2)dx \Rightarrow \int dy = \int{dx} \ + \ \int{xdx} \ + \ y^2*\int{dx} \ + \ y^2*\int{xdx} \Rightarrow y = x + \frac{x^2}{2} + xy^2 + \frac{y^2 x^2}{2} + C$$

Where the heck do I go from here? I can't sepperate the equation, so how do I solve it?

 

[arildno]

This is totally wrong. Do you understand what separation of variables is about?
To help you along a bit, note that your right-hand side may easily be transformed:
$1+x+y^{2}+xy^{2}=(1+x)y^{2}$
Thus, your diff. eq. can be given in the form:
$$y'=(1+x)y^{2}$$

 

[JoshHolloway]

First I would like to say that I wrote the problem very sloppily (i am still learning how to write in the math tex), I think I have fixed it if you want to look at it again.

 

[JoshHolloway]

...your right-hand side may easily be transformed:
$1+x+y^{2}+xy^{2}=(1+x)y^{2}$

I don't understand what you did here.?

 

[JoshHolloway]

Shouldn't $(1+x)y^{2} = y^{2}+xy^{2}$?

Do you mean: $(1+x)+(1+x)y^{2} = 1+x+y^{2}+xy^{2}$

 

[arildno]

Oh, dear, you're right.
The correct identity is:
$$1+x+y^{2}+xy^{2}=(1+x)(1+y^{2})$$
Sorry about that.

 

[arunbg]

The RHS should be transformed into (1+x)(1+y^2).
That's probably what arildno meant to say.

Arun

edit: He's quick to correct himself.

 

[JoshHolloway]

Alright. And then it should go:
$$(1+y^{2})dy=(1+x)dx$$?

 

[arildno]

Again:
Do you understand what separation of variables is about?

 

[JoshHolloway]

What the heck am I doing wrong with the LaTex that I wrote in the first post? Why are the equations all on the same line?

 

[JoshHolloway]

Didn't I just separate variables?

 

[arildno]

Not correctly, anyway.

 

[JoshHolloway]

Oh wait, I should have the reciprocal of (1+y^2) on the left, right?

 

[arildno]

The double slash option for separating lines in Latex is not available here

 

[JoshHolloway]

$$\frac{1}{1+y^{2}}dy=(1+x)dx$$

Is this the correct sepperation of variables?

 

[arildno]

Oh wait, I should have the reciprocal of (1+y^2) on the left, right?

That's right.

 

[JoshHolloway]

$$\tan^{-1}(y) = x + \frac{x^{2}}{2} + C$$

That is supposed to say arctan(y) on the right, I don't know what I did wrong.

 

[arildno]

Try:
$$\tan^{-1}(y)=...$$

 

[JoshHolloway]

Awesome, I prefer to write it that way anyway.

 

[JoshHolloway]

Hey, I have a question about the LaTex. When I go to edit the LaTex, and then resubmit it to post the edit, the edit doesn't show up. It just shows the same thing as before the edit. I have to restart my computer to see the corrections I make. Do you know what I am doing wrong?

 

[JoshHolloway]

Oh, never mind. It seems to work in IE. I am just having the problem with firefox. It must be some setting I have set wrong with it. Thanks for the speedy help though. I really appreciate it.

 

[HallsofIvy]

Hey, I have a question about the LaTex. When I go to edit the LaTex, and then resubmit it to post the edit, the edit doesn't show up. It just shows the same thing as before the edit. I have to restart my computer to see the corrections I make. Do you know what I am doing wrong?

You don't need to "restart" your computer! Just click on the "refresh" button (arrows going in a clockwise circle). Same thing happened to me. Until someone told me about the "refresh", I would copy the corrected text, then DELETE the message and past the corrected text into a new message box!

 

[JoshHolloway]

I know how to refresh it. I have been doing that. And it still doesn't show the correction (in firefox). I even have tried to close firefox and reopen it, and it still doesn't work. One time I even tried to wait a few hours and then refresh the screen and it still didn't work. But it is working OK in IE.