Solving 5th Degree Characteristic Eq. of Linear Homog. Diff. Eq.

[bobmerhebi]

Homework Statement

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 please help. if possible please give a 5th deg eq. thx

 

[Vid]

http://en.wikipedia.org/wiki/Abel–Ruffini_theorem

 

[HallsofIvy]

The characteristic equation is a 5^th 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.

 

[bobmerhebi]

the thing is if I am 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?

 

[Dick]

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.