Linear, non linear and homeogenous and non homeogenous

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SUMMARY

The discussion focuses on the classification of differential equations into linear, non-linear, homogeneous, and non-homogeneous categories. The user seeks clarification on how to demonstrate whether a given equation is linear, specifically in the context of expressing it in the form of a linear differential operator. Additionally, the user inquires about the conditions for homogeneity and how to identify the functions involved. The conversation emphasizes the importance of expressing equations correctly to determine their properties.

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  • Understanding of linear differential equations and their forms.
  • Familiarity with the concepts of homogeneity and non-homogeneity in mathematics.
  • Knowledge of differential operators and their applications.
  • Basic proficiency in mathematical notation and terminology.
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  • Study the properties of linear differential equations and their solutions.
  • Learn how to identify and classify homogeneous and non-homogeneous equations.
  • Explore the method of expressing differential equations in standard forms.
  • Investigate the role of differential operators in solving equations.
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Students and professionals in mathematics, particularly those studying differential equations, as well as educators looking to clarify concepts related to linearity and homogeneity in mathematical contexts.

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Homework Statement


Hi guys, I am having a bit of trouble with this question:

upload_2017-5-3_23-33-20.png


S2. It the linear non linear and homogeneous parts. I think it is a linear equation, as I always think dy/dx (y)=H(x), but is there a way to show this, also for non linear cases. I believe the second part to this question my ans in Inhomeogenous, but once again is there a mathmatical way to show this. I know its not asking for that in the question but for future ref. Also I is omega a constant in this?

Homework Equations

The Attempt at a Solution

 
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For S2 to be linear, we must be able to express it in the form ##Ly=f(t)## for some linear differential operator ##L## and function ##f##. Can you do that?

Re homogeneity: first express the equation in the form given here. Having done that, what are the functions ##M(y,t)## and ##N(y,t)##? Are they both homogeneous?
 

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