I have the IVP:
$$u'(t) = u - t^2 + 1, 0 \le t\le 2, u(0) = 1/2$$
I'm asked to use the Picard-Lindelof thm to show the IVP has a unique solution on $D=\{ (t,u):0\le t\le 2, |u|<10 \}$ and find the best possible Lipschitz constant.
I know the definition for Lipschitz for one variable, but the $t$ is throwing me off here.
So far, I have said:
Let $f(t,u(t)) = u-t^2+1$ and $F_u = 1$, then $f$ is continuous when $u=1/2$ since $f = 3/2-t^2$ is continuous on $t$) and $F_u$ is continuous for all $u$ which implies $f$ is continuous in $t$ . I know I need to show that $f$ is Lipschitz in $u$ but I'm a bit lost. I also don't know how to determine what the "best possible Lipschitz constant" is.
Any help would be great.