Finding solutions to equations of motion

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SUMMARY

The discussion focuses on solving the equation of motion for a critically damped oscillator, represented by the equation x'' + 2yx' + w²x = fe^t, where the natural frequency (w) equals the coefficient of friction (y). The user is tasked with finding a solution for x, specifically Afe^t, where A = f/4. A critical point raised is the inconsistency in the variable "f," which appears to represent different quantities in different contexts, leading to confusion regarding the solution's validity.

PREREQUISITES
  • Understanding of differential equations, specifically second-order linear equations.
  • Familiarity with concepts of critically damped oscillators.
  • Knowledge of the Laplace transform technique for solving differential equations.
  • Basic grasp of dimensional analysis in physics.
NEXT STEPS
  • Study the method of solving second-order linear differential equations with constant coefficients.
  • Learn about the Laplace transform and its application in solving differential equations.
  • Explore the concept of critically damped systems in mechanical engineering.
  • Review dimensional analysis techniques to ensure consistency in physical equations.
USEFUL FOR

Students and professionals in physics and engineering, particularly those focused on dynamics and control systems, will benefit from this discussion.

Ed Quanta
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Ok, so I am dealing with a critically damped oscillator in which the natural frequency(w) of the oscillator is equal to the coefficient of friction (y). I am given the force mfe^t and told to find a solution for x, where

x'' +2yx' +w^2 =fe^t.

How do I go about doing this? The solution that I am supposed to find is Afe^t where A=f/4

I have to solve this for f=mfe^-t also, if this requires a different strategy, let me know I guess.
 
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It would help a lot if you would clarify what you are saying. There is clearly a typo in your equation: it should be
x'' +2yx' +w^2x =fe^t.

But the main problem is that you seem to be using "f" to mean at least two different things. You say "I am supposed to find is Afe^t where A=f/4". Is that f<sup>2</sup>e<sup>t</sup>? But then "I have to solve this for f=mfe^-t". Surely f doesn't mean the same thing on both sides of that equation (since me<sup>-t</sup> is not 0!).
 
I doubt whether the Pro is correct

And what are the dimensions on both sides of the solution
 
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