Question: Determine the largest interval \(\left( a,b \right)\) for which theorem 1 guarantees the existence of a unique solution on \(\left( a,b \right)\) to the given initial value problem

\[y'''-y''+\sqrt{x-1}y=\tan x;\,\,\,y\left( 5 \right)=y'\left( 5 \right)=y''\left( 5 \right)=1\]