FIND THE GENERAL SOLUTION OF THE LINEAR SYSTEM
x'' = 6x + 2y, y'' = 3x + 7y
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.