  The differential Sylvester equation and its symmetric version, the
differential Lyapunov equation, appear in different fields of applied
mathematics like control theory, system theory, and model order reduction. The
few available straight-forward numerical approaches if applied to large-scale
systems come with prohibitively large storage requirements. This shortage
motivates us to summarize and explore existing solution formulas for these
equations. We develop a unifying approach based on the spectral theorem for
normal operators like the Sylvester operator $\mathcal S(X)=AX+XB$ and derive a
formula for its norm using an induced operator norm based on the spectrum of
$A$ and $B$. In view of numerical approximations, we propose an algorithm that
identifies a suitable Krylov subspace using Taylor series and use a projection
to approximate the solution. Numerical results for large-scale differential
Lyapunov equations are presented in the last sections.
