Determining Which Variable Is Independent

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SUMMARY

The discussion centers on identifying independent and dependent variables in the context of differential equations, specifically the equation x^5 y^{(4)} - x^3 y'' + 6y = 0. It is established that the presence of the variable y and its derivatives indicates that y is the dependent variable, while x is typically the independent variable unless otherwise specified. The conversation highlights the importance of understanding the roles of variables in differential equations to accurately interpret solutions.

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  • Understanding of differential equations
  • Knowledge of dependent and independent variables
  • Familiarity with derivatives and their notation
  • Basic concepts of linearity in equations
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  • Study the classification of differential equations, focusing on linear vs. nonlinear types
  • Learn about the role of independent and dependent variables in mathematical modeling
  • Explore the implications of variable selection in differential equations
  • Review examples of differential equations with varying independent variables
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Students and professionals in mathematics, particularly those studying differential equations, as well as educators seeking to clarify the concepts of independent and dependent variables in mathematical contexts.

Bashyboy
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Hello,

I have a question with regards to solving problems that require knowledge of which variable is dependent and which is independent. Suppose I am presented with the differential equation [itex]x^5 y^{(4)} - x^3 y'' + 6y = 0[/itex]; for this differential equation I am asked to determine if it is linear or not. My question does not pertain to linearity, but is, how do I know which variable is the dependent? Is my knowing this solely due to the fact that the equation contains y and its derivatives, or does this not imply that y is the dependent variable?
 
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If there are derivatives of y then it must be dependent on some variable; otherwise what are you differentiating with respect to?

Often the writers are a bit glib in specifying the variable; for example your equation could have x be some arbitrary parameter and y be a function of t, which would change the solutions that you get. Unless specified though if something that looks like a variable (either x or t) appears in the equation then that is typically intended to be the independent variable.
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