Why is general solution of homogeneous equation linear

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Discussion Overview

The discussion revolves around the nature of the general solution of second-order homogeneous differential equations, specifically addressing why it is considered linear. Participants explore the characteristics of linear combinations of solutions and the implications of the notation used in the equations.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about why the general solution, represented as c_1e^(xt) + c_2e^(xt), is considered linear.
  • Another participant clarifies that the general solution is a linear combination of two linearly independent solutions, y1(x) and y2(x).
  • A subsequent reply corrects the notation, suggesting that the exponent should be r instead of x, leading to the expression C_1e^(r_1t) + C_2e^(r_2t).
  • One participant questions whether the linearity of the solution is due to the first power of y_1 and y_2, despite e^rt not being a linear function.
  • Another participant notes that the linearity pertains to y, not x, and references the properties of linear operators.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of linearity in the context of the general solution, with multiple viewpoints presented regarding the nature of the solutions and the notation used.

Contextual Notes

There are unresolved issues regarding the notation and the definitions of linearity in the context of differential equations, particularly concerning the roles of the variables involved.

lonewolf219
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Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...
 
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lonewolf219 said:
Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...

The general solution is a linear combination of two linearly independent solutions y1(x) and y2(x).


p.s. I think something is not right with your notation e^(xt) .
 
Yes, you're right. The x in the exponent of e should be r, where you would find the roots using the characteristic equation. C_1e^(r_1t)+C_2e^(r_2t). So this is a linear solution because y_1 and y_2 are to the first power? Even though the function e^rt is not a linear function?
 
It is linear in y not x.
L is a linear operator if
L[Ʃanyn]=ƩanL[yn]
 
Ah, OK. Thanks guys.
 

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