Solve :
$\displaystyle 4y^{\left( 4 \right)} - 23y^{\left( 2 \right)} - y^{\left( 1 \right)} = 0$
They're messy. You know one right? Since they're all derivatives, then obviously $\displaystyle y=k$ is a solution and that corresponds to factoring out an m from the characteristic equation: $\displaystyle 4m^4-23m^2-m=m(4m^3-23m-1)=0$. Here's the others via Mathematica so please excuse the capital letters which Mathematica uses, and sides, I don't have to type the latex this way .
$\displaystyle \left\{m\to \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}$
$\displaystyle \left\{m\to -\frac{1}{2} \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]+\frac{1}{2} \sqrt{23} \text{Sin}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}$
$\displaystyle \left\{m\to -\frac{1}{2} \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]-\frac{1}{2} \sqrt{23} \text{Sin}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}$
The characteristic equation is:
$\displaystyle 4 \lambda^4 -23 \lambda^2-\lambda=0$
One $\displaystyle \lambda$ factors out to give:
$\displaystyle \lambda ( 4 \lambda^3-23 \lambda -1) =0$
To find the remaining roots (other than $\displaystyle \lambda=0$) we need the roots of:
$\displaystyle 4 \lambda^3-23 \lambda -1 =0$
Which is a depressed cubic and can be solved by Cardarno's trick.
CB