# How to turn a system of first order into a second order

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!

• Not sure this is what you're looking for, but couldn't you take the derivative of the first equation, then substitute $Y$ and $Y^\prime$ in? Apr 20 '15 at 5:58
• Nonlinear version of question: math.stackexchange.com/questions/2175641/… Sep 2 '18 at 7:27

## 1 Answer

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.

• ah, that was much easier than I was making it, just a question how did you get Y, I think that was where I was running into major issues.
– jam
Apr 20 '15 at 6:02
• Im guessing you just solve the first equation correct?
– jam
Apr 20 '15 at 6:03
• I get $Y$ from the first equation $X'=aX+bY$. Apr 20 '15 at 6:03
• Awesome that's what I thought thanks for the help! Storing that trick for later!
– jam
Apr 20 '15 at 6:06