Solving Complicated Autonomous Equations: Tips and Methods

  • Context: Undergrad 
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SUMMARY

This discussion focuses on methods for converting complicated differential equations into autonomous equations. A key technique involves introducing a new variable to eliminate time from the equation, exemplified by transforming the equation (x' - x)t + x = 0 into u' = u by letting u = xt. Additionally, if this method is unsuccessful, participants suggest converting the equation into a system of two autonomous equations by defining a second variable as time.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with autonomous systems
  • Knowledge of variable substitution techniques
  • Basic calculus concepts
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  • Research variable substitution in differential equations
  • Learn about autonomous systems in differential equations
  • Explore methods for converting non-autonomous equations
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Mathematicians, engineering students, and researchers working with differential equations and seeking to simplify complex systems into autonomous forms.

thecoop
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hi guys , i want to write a complicated differential equation as autonomous equations , is there any method ? i have no idea to overcome it !



thank you
 
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It may be possible to introduce a new variable, which will eliminate the time from the equation. For example:

(x' - x)t + x = 0

Let u = xt and the equation becomes

u' = u

If that fails, you can always convert an equation to an autonomous system of two equations in two variables, letting the second variable = time.
 

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