# Thread: Differential equation order 4

1. ## Differential equation order 4

Solve :

$\displaystyle 4y^{\left( 4 \right)} - 23y^{\left( 2 \right)} - y^{\left( 1 \right)} = 0$

2. Originally Posted by dhiab
Solve :

$\displaystyle 4y^{\left( 4 \right)} - 23y^{\left( 2 \right)} - y^{\left( 1 \right)} = 0$
This is a linear constant coefficient ODE and its charteristic equation has four distinct roots, so what is the problem?

CB

3. Originally Posted by CaptainBlack
This is a linear constant coefficient ODE and its charteristic equation has four distinct roots, so what is the problem?

CB
Thank you are you the details?

4. 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 .

$\displaystyle \left\{m\to \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}$

$\displaystyle \left\{m\to -\frac{1}{2} \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]+\frac{1}{2} \sqrt{23} \text{Sin}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}$

$\displaystyle \left\{m\to -\frac{1}{2} \sqrt{\frac{23}{3}} \text{Cos}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]-\frac{1}{2} \sqrt{23} \text{Sin}\left[\frac{1}{3} \text{ArcTan}\left[\frac{2 \sqrt{\frac{3035}{3}}}{3}\right]\right]\right\}$

5. Originally Posted by CaptainBlack
This is a linear constant coefficient ODE and its charteristic equation has four distinct roots, so what is the problem?

CB
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