Solving Characteristic Equation: y'''-y''+y'-y=0

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Discussion Overview

The discussion revolves around solving the characteristic equation y''' - y'' + y' - y = 0, specifically focusing on finding the roots of the equation and the methods for doing so, including factoring and using polynomial division. The scope includes mathematical reasoning and problem-solving techniques related to differential equations.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant expresses difficulty in solving for the complex roots after identifying r = 1 as a real root.
  • Another participant confirms that r = 1 is indeed a root and suggests factoring it out from the characteristic equation.
  • A different participant proposes writing the equation in the form (r - 1)(ar² + br + c) and equating coefficients to solve for the quadratic.
  • Another response reiterates the factoring approach and mentions that the other factor is r² + 1, while also suggesting the use of Wolfram Alpha for factoring.
  • One participant admits to not understanding long division, which is mentioned as a method for polynomial division.
  • A later reply indicates that the participant has understood the approach after receiving assistance.

Areas of Agreement / Disagreement

Participants generally agree on the method of factoring the characteristic equation to find the roots, but there is no consensus on the preferred method of division or the understanding of long division.

Contextual Notes

Some participants express uncertainty regarding the long division method, and there are varying levels of familiarity with the techniques discussed, which may affect the clarity of the solution process.

Who May Find This Useful

Individuals interested in differential equations, particularly those seeking methods for solving characteristic equations and understanding polynomial roots.

newtomath
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I am stuck on solving for the roots of a charactristic equation:

y'''- y''+y'-y=0

where I set r^3-r^2+r-1=0 and factored out r to get r*[ r^2-r +1] -1 =0 to get the real root of 1. How can I solve for the compex roots?
 
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By inspection, r = 1 is a root of your characteristic equation.
In order to find the other roots, you should factor (r - 1) from the char. eq.
 
So what I would do is write:
<br /> (r-1)(ar^{2}+br+c)=r^{3}-r^{2}+r-1<br />
Expand and equate coefficients, then solve the quadratic
 
hunt_mat said:
So what I would do is write:
<br /> (r-1)(ar^{2}+br+c)=r^{3}-r^{2}+r-1<br />
Expand and equate coefficients, then solve the quadratic
Typically people use long division. But in this case it's obvious that the other factor is r^2+1.

Or you could just go to Wolfram Alpha and say "factor r^3-r^2+r-1" :-)
 
Last edited:
I never really understood long division.
 
Got it now, thanks
 

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