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Differential equation problem

  1. Nov 22, 2008 #1
    1. The problem statement, all variables and given/known data

    Solve

    [tex] y \frac{d^2y}{dt^2} + (\frac{dy}{dt})^2 = 1 [/tex]

    2. Relevant equations



    3. The attempt at a solution

    [tex] \frac{dy}{dt} = v[/tex]

    [tex] \frac{d^2y}{dt^2} = v \frac{dv}{dy}[/tex]

    [tex] yv \frac{dv}{dy} + v^2 =1 [/tex]

    [tex] \frac{dv}{dy} + \frac{v}{y} = \frac{1}{vy} [/tex]

    [tex] I(y) = exp ( \int \frac{1}{y} dy)[/tex]

    [tex] I(y) = y [/tex]

    [tex] y \frac{dv}{dy} + v = \frac{1}{v}[/tex]

    [tex] \frac{d}{dy} (vy) = \frac{1}{v}[/tex]

    [tex] y= \frac{ln(v)}{v} + \frac{C}{v} [/tex]

    Thanks.
     
  2. jcsd
  3. Nov 22, 2008 #2
    Hi Ed,
    This is where the error shows up: you are integrating one side with respect to y and the other with respect to v. Notice that your fourth line
    [tex] \frac{dv}{dy} + \frac{v}{y} = \frac{1}{vy} [/tex]
    is not of the form you would normally apply the integrating factor method you have used: the right-hand side should be a function of y only.

    All is not lost, however. From your third line
    [tex] yv \frac{dv}{dy} + v^2 =1 [/tex]
    you can instead use the method of separation of variables.
     
  4. Nov 22, 2008 #3

    Dick

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    Homework Helper

    You are also taking rather of the long way around. The problem is much easier if you notice that your equation is
    [tex]
    \frac{d^2 (y^2/2)}{dt^2} = 1
    [/tex]
     
  5. Nov 22, 2008 #4
    Ok so heres my new attempt :

    [tex] y v \frac{dv}{dy} + v^2 = 1[/tex]


    [tex] y \frac{dv}{dy} + v = \frac{1}{v} [/tex]


    [tex] y \frac{dv}{dy} = \frac{1 - v^2}{v} [/tex]


    [tex] \int \frac{v}{1 - v^2} dv = \int \frac{1}{y} dy [/tex]

    [tex] ln ( \frac{1}{ \sqrt{1 - v^2} }) = ln (y) + C [/tex]

    [tex] y = exp(ln(\frac{1}{\sqrt{1 - v^2}}) + C ) [/tex]

    [tex] y = \frac{1}{\sqrt{1 - v^2}} . e^C [/tex]

    [tex] y \sqrt{1 - v^2} = e^C [/tex]

    I don't think I am getting anywhere with it.

    Thanks for the help.
     
  6. Nov 22, 2008 #5
    Sorry I'm not sure where you got
    [tex]

    \frac{d^2 (y^2/2)}{dt^2} = 1

    [/tex]

    from.
     
  7. Nov 22, 2008 #6

    Dick

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    Homework Helper

    I differentiated y^2/2 twice with respect to t. (y^2/2)'=y*y'. (y*y')'=y*y''+(y')^2. Your equation is y*y''+(y')^2=1.
     
  8. Nov 22, 2008 #7
    I see it now, thanks very much for your help!!!
     
  9. Nov 22, 2008 #8
    Sorry for double-posting but now I'm having doubts if I'm doing the next integral right.

    [tex] \frac{d^2}{dt^2} (\frac{1}{2} y^2) = 1[/tex]

    [tex] \alpha = \frac{1}{2} y^2[/tex]

    [tex] \frac{d \alpha}{dt} = v[/tex]

    [tex] \frac{dv}{dt} = 1[/tex]

    [tex] v = t + C [/tex]

    [tex] \frac{d \alpha}{dt} = t + C[/tex]

    [tex] \alpha = \frac{1}{2} t^2 + Ct[/tex]

    [tex] y^2 = t^2 + Ct [/tex]

    [tex] y = \sqrt{t^2 + Ct} [/tex]

    Correct?
     
  10. Nov 22, 2008 #9

    Dick

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    Just integrate the first equation twice. The first time you get d(y^2/2)/dt=t+C. The second time you get y^2/2=t^2/2+Ct+D. So y^2=t^2+Ct+D. (I didn't multiply C and D by 2 since they are just constants anyway). So your answer is right. But you should have picked up another constant when you integrated dalpha/dt.
     
  11. Nov 22, 2008 #10
    This wasn't wrong (although technically the C is -C). You can solve your last equation for v, which is dy/dt, and then separate variables again to finish the job (taking care to remember there are two square roots). Just a general note (Dick will go down the other track with you), you should expect two arbitrary constants in your solution.
     
  12. Nov 22, 2008 #11
    Cool, got it now.

    Thanks again!!
     
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