  A broad set of sufficient conditions consisting of systems of linear partial
differential equations is presented which guarantees that the Wronskian
determinant solves the Korteweg-de Vries equation in the bilinear form. A
systematical analysis is made for solving the resultant linear systems of
second-order and third-order partial differential equations, along with
solution formulas for their representative systems. The key technique is to
apply variation of parameters in solving the involved non-homogeneous partial
differential equations. The obtained solution formulas provide us with a
comprehensive approach to construct the existing solutions and many new
solutions including rational solutions, solitons, positons, negatons,
breathers, complexitons and interaction solutions of the Korteweg-de Vries
equation.
