Linear vs nonlinear diff equation II

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Discussion Overview

The discussion focuses on the distinctions between linear and nonlinear differential equations, particularly regarding the nature of coefficients and the interpretation of constants and functions within these equations. Participants explore definitions and implications of terms used in the context of differential equations.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the definition of constants in the context of linear differential equations, questioning how a function can be a constant.
  • Another participant clarifies that coefficients can be either functions of x or constants, but not both, suggesting an exclusive interpretation of "or".
  • A participant proposes that a constant function can be viewed as a trivial case of a function of x, comparing it to a machine that outputs the same value regardless of input.
  • There is a discussion about the meaning of "trivial solution," with some participants suggesting it refers to the zero solution, while others note it can also imply a lack of interest.
  • A later reply emphasizes that in linear equations, coefficients cannot depend on y or its derivatives, highlighting a key characteristic of linearity.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of "trivial solution" and its implications, and there is some disagreement regarding the definitions and characteristics of linear versus nonlinear equations.

Contextual Notes

There are unresolved assumptions regarding the definitions of terms like "trivial solution" and the implications of coefficients being constants versus functions. The discussion also reflects varying interpretations of the language used in the context of differential equations.

Calpalned
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For the linear differential equation, it says that a_n and b_n are constants or functions of x. This implies that a function of x = f(x) = constant. How can a number be a constant if it is a function? I think I'm misunderstanding something.
 

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They mean that ## a_n ## and ## b_n ## can be either functions of ## x ## or constants, but not both. In other words, they are using "or" in the exclusive sense, not in the inclusive sense. We could also think of ## f(x) = ## constant as a special case of a "function of ##x##", albeit a trivial one. Think of a machine that spits out the same thing, no matter what you feed into it.
 
Geofleur said:
They mean that ## a_n ## and ## b_n ## can be either functions of ## x ## or constants, but not both. In other words, they are using "or" in the exclusive sense, not in the inclusive sense. We could also think of ## f(x) = ## constant as a special case of a "function of ##x##", albeit a trivial one. Think of a machine that spits out the same thing, no matter what you feed into it.
If I am not mistaken, a trivial solution is one where the solution is zero?
 
Well, trivial can also mean "not very interesting".
 
Calpalned said:
If I am not mistaken, a trivial solution is one where the solution is zero?
Yes, the trivial solution is the zero solution.
 
The crucial point in "linear" equations is that the coefficients, [itex]a_i[/itex], cannot be functions of y or any of its derivatives.
 

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