Existence, Uniqueness & Bifurcations · Hard problem

Two solutions of $y' = t\sin y + \cos t$ start at $t = 0$ with values differing by $10^{-3}$. Let $L(t)$ be the smallest bound on $\left|\dfrac{\partial f}{\partial y}(t, y)\right|$ over all $y$, at each time $t \ge 0$. Gronwall's inequality gives $|y_1(t) - y_2(t)| \le |y_1(0) - y_2(0)|\exp\!\left(\displaystyle\int_0^t L(s)\,ds\right)$. What upper bound does this give for $|y_1(t) - y_2(t)|$ at $t = 2\sqrt{\ln 10}$? Express your answer as a common fraction.