Solution to x + 1/(1 + x)^2 = 1/x^2


by jdstokes
Tags: 1 or 1, 1 or x2, solution
jdstokes
jdstokes is offline
#1
Dec5-05, 08:07 PM
P: 527
In other words

x^5 + 2x^4 + x^3 - 2x - 1 = 0.

I am aware that fifth order polynomials are generally not analytically soluble. Are there any clever ways to at least approximately solve this equation without resorting to numerical methods or fourth order taylor approximation which does not capture the asymptotic behaviour.
Phys.Org News Partner Science news on Phys.org
Lemurs match scent of a friend to sound of her voice
Repeated self-healing now possible in composite materials
'Heartbleed' fix may slow Web performance
Hurkyl
Hurkyl is offline
#2
Dec5-05, 09:06 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
I am aware that fifth order polynomials are generally not analytically soluble.
That's not quite accurate:

There is no (general) expression for the roots of a polynomial of degree 5 or higher in terms of the integers, +, -, *, /, and n-th root functions. (for integer n)


There are certainly analytic solutions -- e.g. there are functions that maps 6 complex numbers to the solution to a polynomial with those numbers as coefficients, and I believe they can be made analytic on large regions.

I also think that such things can be solved in terms of sines and cosines (and arcsines and arccosines), but I don't know how much that helps, since generally sines and cosines can only be "evaluated" through numerical approximation.
devious_
devious_ is offline
#3
Dec6-05, 10:37 PM
P: 347
That's the first time I've seen a quintuple post.

jdstokes
jdstokes is offline
#4
Dec7-05, 08:02 AM
P: 527

Solution to x + 1/(1 + x)^2 = 1/x^2


lol. I've heard about such solutions, they typically span hundreds of pages and are thus of little practical use. Any other thoughts on approximate solutions, or am I stuck up the proverbial creek?
Werg22
Werg22 is offline
#5
Dec7-05, 09:28 PM
P: 1,520
Actually you could use Newton's method altought with such a function it would take some time.


Register to reply

Related Discussions
How to determine if the mixing of two solution would result in buffer solution? Biology, Chemistry & Other Homework 3
pH of solution Biology, Chemistry & Other Homework 1
Draft paper deriving a non-empty, stationary, axisymmetric solution solution of Einstein's Equations, based on the Lorentz Force Law General Physics 15
Particular solution DE Calculus & Beyond Homework 6
[SOLVED] Draft paper deriving a non-empty, stationary, axisymmetric solution solution General Physics 16