Some ODEs contain processes with very different time scales. A system is called stiff when numerical stability forces an explicit method to use steps much smaller than the time scale of the behavior of interest.

For the test equation

$$y'=\lambda y,$$

explicit Euler gives

$$y_{n+1}=(1+h\lambda)y_n.$$

Even when the true solution decays, the numerical solution becomes unstable if $h$ is too large relative to a large negative $\lambda$.

Implicit Euler instead uses

$$y_{n+1}=y_n+h f(t_{n+1},y_{n+1}),$$

so the new state appears inside the derivative evaluation and must generally be solved for.

Implicit methods require more work per step but can remain stable for much larger steps on stiff problems. Stability properties can therefore matter more than formal order when choosing a time integrator.