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Higher-Order DE factoring

  • #1
Hi,

Can somebody please help me to factor the following DE?

[tex]2\frac{d^5y}{dx^5} -7\frac{d^4y}{dx^4} + 12\frac{d^3y}{dx^3} + 8\frac{d^2y}{dx^2} = 0[/tex]

The auxiliary equation of above DE is

[tex]2m^5 - 7m^4 + 12m^3 + 8m^2 = 0[/tex]

[tex]m^2(2m^3-7m^2 + 12m + 8) = 0[/tex]

The equation of [tex]2m^3-7m^2+12m+8[/tex] is cannot be factored in any form, at least after I try for several times.

Thanks in advance
 

Answers and Replies

  • #2
The equation of [tex]2m^3-7m^2+12m+8[/tex] is cannot be factored in any form, at least after I try for several times.
That polynomial can certainly be factorized. You just need to look a little harder, or try some basic root finding
 
  • #3
Hi Scottie,

Thanks in advance.

Since the degree of polynomial is > 2 then to find the factor is “by division”.

I’ve try that way but there is no solution.

Do you mean that “basic root finding” is[tex]\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]?

If so, since it’s only can be used for 2-degree polynomial, how to use that formula in 3 degree of polynomial equation?

Thank you very much
 
  • #4
epenguin
Homework Helper
Gold Member
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701
Just plot
[tex](2m^3-7m^2 + 12m + 8) [/tex]
with a plotting hand calculator or computer plotting routine or by hand calculation. You will see a root, then check you have an exact one.

Note you have already factored the equations somewhat; that gives you some solution already which will later enter into the general solution.
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,793
922
The "rational root theorem" says that the only possible rational roots must have denominator divisible by 2 (coefficient of x3) and numerator a factor of 8. That is, the only possible rational roots are [itex]\pm 1/2[/itex], [itex]\pm[/itex]1, [itex]\pm[/itex]2, [itex]\pm[/itex]4, or [itex]\pm[/itex]8. Since you can get reasonable factoring only with rational roots, try those and see if any are roots.
 
  • #6
Graphing 2m3-7m2+12m+8 reveals that [tex]\frac{-1}{2}[/tex] is a root.

Using polynomial long division by (m+[tex]\frac{1}{2}[/tex]) yields this:

(m+[tex]\frac{1}{2}[/tex])(2m2-8m+16)=2m3-7m2+12m+8

See if you can finish from here.
 

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