Solving ODEs by Elimination

1. Apr 27, 2013

Dear Fellows, I want to solve 4 differential homogeneous equations for non trivial solution. I have found two methods
1) Determinent. 2) Elimination
By det. I got 8th order resultant equation and by second method I got 10th order. As each of initial equation is 2nd order so 8th is correct. But what goes wrong in Elimination I could not understand.
Example.
x+2Dy+D2z=0
9Dx+4y-6D3z=0
x+D2+Dz=0

determinant gives 5th order equation
while if we eliminate by taking value of z from eq i and put in ii and iii, and we will get iv , v. which finally gives 4th order equation.

I doubt in taking the value of z from first equation, is this thing allowed or not? if no then why?