It's often found in books that the global truncation error of the forward Euler method applied to $\dot{y} (t) = f(t, y(t))$ is given by something like
$$ \frac{\exp(LT) -1}{L} \frac{Mh}{2},$$
with $L$ the Lipshitz constant of $f$, $h$ the step length and $M$ a bound on the second derivative of $y$. I was wondering: apart from solving the stability issue, does backward Euler method improve this exponential dependence on the Lipshitz constant?