The solution to this differential equation became complex at t=1. Without Mathematica, a user would have to anticipate this before solving the problem. Technology Guide: Automatic algorithm selection, dynamic type detection The Mathematica Book: Sections 2.1.1, 3.1.1

Computations in purely numerical systems often fail because their number representation is insufficient for the task--for example when IEEE floating-point overflow or underflow occurs, or when complex numbers appear in a floating-point computation.

With dynamic type switching (DTS), Mathematica detects these problems and switches number systems without user interaction. Mathematica can therefore often return accurate results in situations where non-Mathematica, numerical-only systems fail.

An example of DTS is found when using the numerical differential equation solver NDSolve. An initially real solution might become complex during the calculation, automatically triggering Mathematica's just-in-time (JIT) compiler to recompile optimized byte code.