Question

Hello! Is it posible that you find particular solution in nonhomogeneous DE 2 order with:

Answer 1

Wronskian or with

c1'*y1+c2'y2=0
c1'*y1'+c2'*y2'=g(x)

Answer 2

You've witten down equations from the middle of a procedure for solving a system of linear ODEs, which are in turn a way to analyze second order inhomogeneous ODEs.

So if you want to understand these equations, I'd put them back in that context. Have a look in your textbook.

Alternatively, this might be helpful:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-28-matrix-methods-for-inhomogeneous-systems/