Summary: "In a Banach space E, the Cauchy problem v'(t)+A(t)v(t)=f(t)quad (0leqslant tleqslant 1), v(0)=v_0 is considered for a differential equation with linear strongly positive operator A(t) such that its domain D=D(A(t)) is everywhere dense in E independently off t and A(t) generates an analytic semigroup exp{-sA(t)} (sgeqslant 0). Under some natural assumptions on A(t), we establish coercive solvability of the Cauchy problem in the Banach space C^{beta,gamma}_0(E). We prove a stronger estimate of the solution compared to estimates known earlier, using weaker restrictions on f(t) and v_0."