Understanding Autonomous Functions: Helping Me Grasp It!

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SUMMARY

The discussion focuses on the concept of autonomous functions in the context of differential equations. An autonomous function is defined as one that does not explicitly depend on the independent variable, as illustrated by the example of the function x' = x^3, which is autonomous, versus x' = t x^3, which is not due to its explicit dependence on the variable t. The continuity of the derivative f' and the domain of the function being all real numbers (R) are crucial factors in determining autonomy. Understanding these distinctions is essential for grasping the fundamentals of autonomous differential equations.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with the concept of continuity in calculus
  • Knowledge of function derivatives
  • Basic understanding of independent and dependent variables
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  • Study the properties of autonomous differential equations
  • Learn about the implications of explicit versus implicit dependence in functions
  • Explore examples of non-autonomous functions and their characteristics
  • Investigate the role of initial conditions in solving differential equations
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Students and educators in mathematics, particularly those focusing on calculus and differential equations, as well as anyone seeking to deepen their understanding of autonomous functions and their applications.

simo1
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I do not think I understand fully the concept of determining whether a function is autonomous> may you please help me understnad
eg I was given thsi function
x' = x^3 x(1)=1
i said

f(z)=z^3
where f'(z) = 3x^2 and the domain of f is in all R where the domain of f is also in all R hence f' is contionous then it is autonous. but I don't even know why we do this
 
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Re: autonomous DE

Examples to compare may help: $$\begin{cases} x' = x^3 & \text{is autonomous,} \\ x' = t x^3 & \text{is not.} \end{cases}$$ In the second case we have an explicit dependence on the independent variable in the differential equation.
 

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