Is the D.E. Solution Affected by the Presence of a Non-Homogeneous Term?

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Homework Help Overview

The discussion revolves around the characteristics of a differential equation of the form ay'' + by' + cy = R(x) and how the presence of the non-homogeneous term R(x) affects the damping classification of the system (underdamped, critically damped, or overdamped). Participants are exploring the relationship between the damping conditions defined by the coefficients and the role of R(x).

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning whether the classification of damping is influenced by the non-homogeneous term R(x). Some express uncertainty about the implications of the definitions provided in their texts.

Discussion Status

There is an ongoing exploration of the definitions of damping and their applicability to the given differential equation. Some participants suggest reviewing the definitions, while others reference external resources to clarify the relationship between the homogeneous equation and damping.

Contextual Notes

One participant notes that the definition of damping in their text does not explicitly state the relevance of R(x), leading to some confusion about its role in determining the damping condition.

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Homework Statement



Let ay'' + by' +cy = R(x)

Determining whether a system is under/over/critically damped depends on the size of b^2 compared to 4ac.

Does it depend at all on R(x)?


Homework Equations



Characteristic equation, quadratic equation.

The Attempt at a Solution



I've found a general solution to a D.E. where b^2 < 4ac, and is therefore underdamped (if the D.E = 0)

However, I'm not sure if I can still conclude that the system is underdamped if the D.E. = R(x).
 
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Look after the definition of being under/over/critically damped.

ehild
 
Darkmisc said:

Homework Statement



Let ay'' + by' +cy = R(x)

Determining whether a system is under/over/critically damped depends on the size of b^2 compared to 4ac.

Does it depend at all on R(x)?
No, it does not. As ehild suggests, surely the definition of "under/over/critically damped" is given in your text?


Homework Equations



Characteristic equation, quadratic equation.

The Attempt at a Solution



I've found a general solution to a D.E. where b^2 < 4ac, and is therefore underdamped (if the D.E = 0)

However, I'm not sure if I can still conclude that the system is underdamped if the D.E. = R(x).
 
Thanks. The definition only referred to b^2 - 4ac without explicitly stating that R(x) was irrelevant to damping.
 

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