solutin exists and is unique

[sanaspek]

Show that the solution u(t) of u' = 1/(t^2 + e^u); u(t0) = u0 exists
and is unique for all t >= t0.
(Can you prove the same conclusion for t > min(t0; 0)
(b) Show that limt u(t) with t--> infinity exists.

for a) i did this:

Let f(t; u) = 1/(t^2 + e^u). Function f is continuous as composition
of continuous elementary functions and the denominator is positive.
Thus, ??????-I do not remember the name of the (please help me) theorem???,
solution exists.
Let us check if f satises Lipschitz condition with respect to u.
f' = -(e^u)/(t^2 + e^u)^2
{ the derivative exists as the denominator is positive.
|f'|=|-(e^u)/(t^2 + e^u)^2|<=1/(t_0)^2

Thus, f is Lipschitz continuous with constant 1/(t_0)^2. Picards Theorem
gives uniqueness of the solution.
If t > min(t0; 0)?????-I do not know
I gess we should use peano theorem... But I do not know how to show that
the solution exists too.

b) I am not sure if
i gess we need to show that u(t) >0, decreases and bounded.
but I do not know how to show thet.

Help me please.

thank You!!!

 

[Deleted member 4993]

Show that the solution u(t) of u' = 1/(t^2 + e^u); u(t0) = u0 exists and is unique for all
t >= t0: (Can you prove the same conclusion for t > min(t0; 0) and WHY?
(b) Show that lim u(t) with t-->infinity exists.

Did you try to apply Cauchy-Picard existence theorem?

Please share with us your work, indicating exactly where you are stuck - so that we know where to begin to help you.