Wanting an explanation to the solution method of the system of differential equations.

Wanting an explanation to the solution method of the system of
differential equations.

Please can anybody help me with explaining, in detail, why and how the
author solves the system of differential equations by such a step (see
below)? How the equation 6.3 is formed step by step? Why such a column
basis is valid? It would be extremely helpful if someone knows the name of
the method or could give an exemplification.