X89codered89X
Jan29-12, 06:37 PM
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.
{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.