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Help with a diff eq

  1. Aug 29, 2007 #1
    Never expected to be pleading for help so soon, and especially not on a differential equation, which I usually am good at. But for whatever reason, I cannot solve this problem:

    y*d(y,x,2) + (d(y,x))^2 + 1 = 0

    Any help would be greatly appreciated!!

    ETA: I know I'm supposed to substitute u=d(y,x) and u*d(u,y)=d(y,x,2) but I can't get any farther than that.

    [tex]
    0 = y \frac {d^2y} {dx^2} + (\frac {dy} {dx})^2 + 1
    [/tex]
     
    Last edited: Aug 29, 2007
  2. jcsd
  3. Aug 29, 2007 #2
  4. Aug 29, 2007 #3
    Sorry, yeah it's an ODE. I know the notation is a little weird but it's the easiest way for me to type it.
     
  5. Aug 29, 2007 #4

    Dick

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    I think what you want to notice is that (y*y')'=y''*y+y'*y'. So you want u=y*y'. In terms of u you have a first order ode. Once you've solved for u, then it's separable.
     
  6. Aug 29, 2007 #5
    Yeah, but I can't separate it! Or rather, I can separate it, but I get completely unworkable results. It tends to fall apart when I get to

    -ln(u^2 + 1)/2 = ln(y) + C
     
  7. Aug 29, 2007 #6

    Dick

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    With this substitution the ode becomes u'+1=0. Can you solve that? I think you can.
     
  8. Aug 29, 2007 #7
    Ohh, okay, I see how that's different than the substitution I was using. Let me try this again (again).
     
  9. Aug 29, 2007 #8
    Success! [tex] y = \sqrt{-x^2 + x + 1} [/tex]

    Thank you SO much! No one in my class has been able to get that, we've been frantically IMing back and forth all night.
     
  10. Aug 29, 2007 #9

    Dick

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    Sorry to rain on your parade, but you still haven't got it. Aside from the fact it's simply wrong, a second order ode should have two undetermined constants. Where are they? I think you know the general pattern of the solution. Try and do it again, carefully this time.
     
  11. Aug 29, 2007 #10
    I guess I didn't mention that I was given two BC ( y(1)=1 and y'(1)=0 ) and I was able to solve for them. And my final answer checks out. Phew.
     
  12. Aug 29, 2007 #11

    Dick

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    Ok. Guess that works.
     
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