Higher-Order Differential Equations

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Homework Help Overview

The discussion revolves around finding the general solution to a higher-order differential equation, specifically a third-order linear homogeneous equation with constant coefficients.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of an auxiliary equation to find solutions, with one participant confirming the factorization of the left side of the equation. Questions arise regarding the application of the quadratic formula to determine additional solutions.

Discussion Status

The discussion is active, with participants exploring the factorization of the auxiliary equation and confirming the approach to find the remaining solutions. There is a focus on the forms of the solutions, but no consensus on the specific values or constants has been reached.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information they can share or the methods they can use.

recon_ind
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Homework Statement



Find the general solution of the given higher-order differential equation.

d3x/(dt3) - d2x/(dt2) - 4x = 0

Homework Equations



Use an auxiliary equation such as m3 - m2 - 4 = 0

The Attempt at a Solution



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That is the auxiliary equation you want to use. As it turns out, the left side can be factored, yielding (m - 2)(m2 + m + 2) = 0

There are three distinct solutions, so the solution to this homogeneous problem will be x(t) = c1e2t + c2em2t + c3em3t. All you have to do is find the other two constants, m2 and m3.
 
So, I should just use the quadratic formula to find the other two solutions?

Thanks man.
 
The other two solutions are of the form [tex]\alpha[/tex] [tex]\pm[/tex] [tex]\beta[/tex]
 
recon_ind said:
So, I should just use the quadratic formula to find the other two solutions?
Yes. And yes, the two solutions are of the form a +/- b. One of your values of m will be a + b, and the other will be a - b.
 

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