# Third order DE

• Apr 30th 2011, 11:09 AM
Naples
Third order DE
$\displaystyle 2y^(^3^) + 9y" + 12y' + 5y =0$
Tried using an auxiliary equation:
$\displaystyle 2m^3 + 9m^2 + 12m + 5 = 0$

Not sure how to factor that so I don't know where to go from there, I feel like I might be missing something obvious, any help is appreciated
• Apr 30th 2011, 11:18 AM
TheEmptySet
Quote:

Originally Posted by Naples
$\displaystyle 2y^(^3^) + 9y" + 12y' + 5y =0$
Tried using an auxiliary equation:
$\displaystyle 2m^3 + 9m^2 + 12m + 5 = 0$

Not sure how to factor that so I don't know where to go from there, I feel like I might be missing something obvious, any help is appreciated

You need to use the rational root theorem.

Since all the terms are positive the roots must be negative and by the rational root theorem the only possible roots are

$\displaystyle -1,-5,-\frac{1}{2},-\frac{5}{2}$

If you check negative one you will find that it is a root so $\displaystyle x+1$ is a factor by long division you get

$\displaystyle (m+1)(2m^2+7m+5)$