Define u= x' and v= y'. Then the differential equation x"= 6x+ 2y becomes u'= 6x+ 2y and y"= 3x+ 7y becomes v'= 3x+ 7y.

So instead of two second order equations we have four first order equations:

x'= u, y'= v, u'= 6x+ 2y, and v'= 3x+ 7y which we could also write as the single first order matrix equation:

And the first step in that is to find the eigenvalues and eigenvectors of the coefficient matrix.