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Help with nonlinear 1st order ODE

  1. Jan 29, 2012 #1
    So I'm supposed to prove that
    [itex]{x}^{.}(t) = x^{2}+ t^{2}[/itex] with [itex] x(0) = 0 [/itex] blows up before [itex] t = 1 [/itex].

    I'm not sure what method to use to solve I've tried setting up an integral such as [itex]\int^{x(t)}_{x(0)} \frac{dx}{x^{2}+t^{2}} = \int^{t}_{0} dt[/itex] but I didn't think I could do this since 't' is varying over time on the Left Hand side and I'm integrating with respect to x.

    The only other clue I have is to use a comparison ODE, which was mentioned in class, in which would use a function, say... [itex] g^{.}(t) =< x^{.}(t) [/itex] which was easier to work with. If I were able to prove that the lesser function exploded before [itex] t= 1 [/itex], then logically the greater one explodes. The thing is I don't know what function I would even chose to set this up. Any ideas?

    ..and Thank youuuuu.
     
    Last edited: Jan 29, 2012
  2. jcsd
  3. Jan 30, 2012 #2
    Hello !

    The ODE is on the Riccati kind. First let x(t) = -y'/y
    leading to y''+t²y=0
    which is on the Bessel kind
    y(t) = sqrt(t)*(c1*J(k,X)+c2*J(-k,X) )
    J(k,X) is the Bessel function of order k
    k=1/4
    X=t²/2
    k=1/4
     
  4. Jan 30, 2012 #3
    Right OK!! can I then substitute that back in somehow to show that it explodes at t -> 1? I don't necessarily need to get it solved. Just know how it behaves.
     
  5. Jan 30, 2012 #4
    From my results it doesn't "explodes" close to t=1, but at t=2.003147 which is the first root of the BesselJ function of order -1/4.
     

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