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Nonlinear First Order Differential Equations

  • #1
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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:
[tex]y\prime=1+x+y^2+xy^2[/tex]

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:
[tex]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[/tex]

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

Answers and Replies

  • #2
arildno
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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:
[itex]1+x+y^{2}+xy^{2}=(1+x)y^{2}[/itex]
Thus, your diff. eq. can be given in the form:
[tex]y'=(1+x)y^{2}[/tex]
 
  • #3
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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.
 
  • #4
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arildno said:
...your right-hand side may easily be transformed:
[itex]1+x+y^{2}+xy^{2}=(1+x)y^{2}[/itex]
I don't understand what you did here.?
 
  • #5
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Shouldn't [itex](1+x)y^{2} = y^{2}+xy^{2}[/itex]?

Do you mean: [itex](1+x)+(1+x)y^{2} = 1+x+y^{2}+xy^{2}[/itex]
 
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  • #6
arildno
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Oh, dear, you're right.
The correct identity is:
[tex]1+x+y^{2}+xy^{2}=(1+x)(1+y^{2})[/tex]
Sorry about that.
 
  • #7
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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.
 
  • #8
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Alright. And then it should go:
[tex](1+y^{2})dy=(1+x)dx[/tex]?
 
  • #9
arildno
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Again:
Do you understand what separation of variables is about?
 
  • #10
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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?
 
  • #11
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Didn't I just seperate variables?
 
  • #12
arildno
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Not correctly, anyway.
 
  • #13
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Oh wait, I should have the reciprocal of (1+y^2) on the left, right?
 
  • #14
arildno
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The double slash option for separating lines in Latex is not available here
 
  • #15
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[tex]\frac{1}{1+y^{2}}dy=(1+x)dx[/tex]

Is this the correct sepperation of variables?
 
  • #16
arildno
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JoshHolloway said:
Oh wait, I should have the reciprocal of (1+y^2) on the left, right?
That's right. :smile:
 
  • #17
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[tex]\tan^{-1}(y) = x + \frac{x^{2}}{2} + C[/tex]

That is supposed to say arctan(y) on the right, I don't know what I did wrong.
 
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  • #18
arildno
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Try:
[tex]\tan^{-1}(y)=...[/tex]
 
  • #19
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Awesome, I prefer to write it that way anyway.
 
  • #20
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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?
 
  • #21
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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.
 
  • #22
HallsofIvy
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JoshHolloway said:
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!
 
  • #23
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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.
 

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