TITLE: How to turn a system of first order into a second order
QUESTION [5 upvotes]: So I have two equations 
$X' = aX + bY$
$Y' = cX + dY$
I want to convert it back to a second order equation with the form 
$X'' + \alpha X' + \beta X$ with $\alpha,\beta$ in terms of a,b,c,d. 
I have been racking my brain for hours trying to go backwards from a reduction of order, but just can't seem to figure it out. Any help would be much appreciated!

REPLY [6 votes]: If you derive the first equation, you get: 
$$  X ^"=aX '+bY ' $$ 
(if you are considering $a$ and $b$ as constants).
But we have $Y'=cX+dY $, so substitute in the above equation, you get 
$$ X ^" = a X' +b(cX+dY).$$
Note that $Y=\frac{1}{b}(X'-aX)$ for $b \neq 0$.
So, substituting again you get the final answer.