I want to estimate the difference $|x(T)-y(T)|,T=0.5$, where $$x'(t)=sin(tx)$$ $$y'(t)=ty$$ With $x(0)=y(0)=0.2$
My approach $|x(T)-y(T)|=|0+\int_0^T sin(sx(s))-sy(s) ds|\leq$ $T\cdot max_{s\in [0,T]}(|sin(s\cdot x(s))-s\cdot y(s)|)$
Any ideas on how to continue to get a good estimate? A trivial way led me to: $diff \leq 0.5$ which feels not good enough.