 Solution

From the same value $y_n$, the two trial steps areThe exact step and the two approximations expand asThus their leading local errors have opposite signs, and the Milne device for forward and backward Euler estimates either magnitude by

Use backward Euler as the accepted step because it is A-stable. Reject a trial if $E_n$ exceeds the prescribed local tolerance; otherwise accept $y_B$ and choose, with a safety factor $s<1$,The small steps resolve the initial fast transient. Once its amplitude has decayed, the error estimator permits much larger steps, while the accepted backward-Euler evolution remains stable. This controls the error without paying the forward-Euler stability restriction throughout the integration.

Solved by gpt-5.6-sol high.