Help with nonlinear 1st order ODE

[X89codered89X]

So I'm supposed to prove that
${x}^{.}(t) = x^{2}+ t^{2}$ with $x(0) = 0$ blows up before $t = 1$.

I'm not sure what method to use to solve I've tried setting up an integral such as $\int^{x(t)}_{x(0)} \frac{dx}{x^{2}+t^{2}} = \int^{t}_{0} dt$ 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... $g^{.}(t) =< x^{.}(t)$ which was easier to work with. If I were able to prove that the lesser function exploded before $t= 1$, 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.

 

[JJacquelin]

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

 

[X89codered89X]

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.

 

[JJacquelin]

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.