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For rigor, you might try to ask on Math.SE or MO.SE. In this answer, we will give a heuristic derivation via discretization. We will use a slightly different notation to connect to usual time-evolution in QM, but the idea is the same: \begin{align} U_{\lambda}(t_f,t_i)~=~&T e^{-\frac{i}{\hbar}\int_{t_i}^{t_f}\! dt~ H_{\lambda}(t)}\cr ~=~&\lim_{N\... 0 T is the ordered product of many infinitesimal factors of V(x,t) = \exp\{\epsilon U(x,t)\} \approx 1+ \epsilon U(x,t). $$Apply Liebniz' product rule to differentiate. If you insist on a rigorous derivation then some properties of the operator U(x,t) need to be adduced. 1 Such analytical expressions were derived for special cases by multiple authors. More general results were obtained by Yang and Yang in C. N. Yang and C. P. Yang One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System, Phys. Rev. 150, 327 (1966). For example, the ground ... 0 A single autonomous (possibly nonlinear) 2nd-order ODE$$F(x,\dot{x},\ddot{x})~=~0. \tag{1}$$can in principle be written as a couple of autonomous (possibly nonlinear) 1st-order ODEs of the form$$\dot{x}~=~f(x,y), \qquad \dot{y}~=~g(x,y). \tag{2} One may show that there always exists an integral of motion/first integral for the latter system (2), at ...