Help with nonlinear 1st order ODE

  • #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:

Answers and Replies

  • #2
798
34
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
 
  • #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.
 
  • #4
798
34
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.

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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