# Homework Help: Help in: finding the roots of the characteristic eq. of an nth order differential eq.

1. Nov 21, 2008

### bobmerhebi

1. The problem statement, all variables and given/known data

Given an nth order linear homog. diff eq.

how can I find the solution for its nth degree characteristic eq?

I know its simple Algebra but plz help. if possible plz give a 5th deg eq. thx

2. Nov 21, 2008

### Vid

3. Nov 21, 2008

### HallsofIvy

Re: help in: finding the roots of the characteristic eq. of an nth order differential

The characteristic equation is a 5th degree polynomial equation. As the link Vid provided says, there is NO general formula for solving fifth or higher degree polymnomial equations. You can try, for example, the "rational root theorem" which says that if m/n is a rational number satifying $a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_0=$, with all coefficients integer then m must evenly divide $a_n$ and n must evenly divide $a_0$. IF there is a rational root, that at least reduce the possiblilities.

4. Nov 21, 2008

### bobmerhebi

Re: help in: finding the roots of the characteristic eq. of an nth order differential

the thing is if im given an ordinary linear homog. differe eq of order greater than 2. how should i solve it?

how could i get the roots of the eq. LaTeX Code: a_nx^n+ a_{n-1}x^{n-1}+ \\cdot\\cdot\\cdot+ a_0= 0
in order to find the general sol. of the D.E.

forexample: 4y''' - 3y' + y = 0. this eq. gives 4m3 - 3m + = 0 as a characteristic eq.. how can this low deg. eq be solved? & how can it be applied to higher degree ones. (in other words i need a fast & easy way to find the roots to use in sovlving D.E.'s).

I get that the roots of the above eq. are -1 & 1/2; where 1/2 is a repeated root.
i got the roots by looking @ the divisors of the 1st & last coefficients & dividing the divisors of the last coefficient by the divisors of the 1st one (like +or - 1/4, +/- 1/2, ...) & them checking which makes the polynomial zero. then simplifying the polynomial by dividing it by (x - root found) & finding a low deg polynomial.

but this way takes time, specially if i have a higher order D.E.

so what do u have to say?

5. Nov 21, 2008

### Dick

Re: help in: finding the roots of the characteristic eq. of an nth order differential

Do what you did. Factor them. If you can't factor them, you are in big trouble as far as solving them exactly. You can always solve them numerically. I'm sorry if it takes time. But that's life.