Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrodinger equation.
The Wei-Norman formula locally relates the Magnus solution of a system of linear time-varying ODE with the solution expressed in terms of products of exponentials by means of a set of nonlinear differential equations in the parameters of the two types of solutions. A closed form expression of such formula is proposed, based only on the structure constants of the Lie algebra generated by the ODEs.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.