Solutions of the differential equation for the over-damped case

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SUMMARY

The discussion focuses on solving the differential equation for the over-damped case, specifically when the condition ω < (1/2τ) is met. The solution involves the expression λ = -(1/2τ) ± √[(1/4π^2) - (ω^2)], highlighting the critical point when ω equals (1/2τ), where the discriminant becomes zero. Participants seek clarification on the solution process for values of ω that are less than (1/2τ), indicating a need for a detailed explanation of the differential equation involved.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with over-damped systems in physics
  • Knowledge of the parameters τ and ω
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the general form of second-order differential equations
  • Learn about the characteristics of over-damped, under-damped, and critically damped systems
  • Explore the implications of the discriminant in quadratic equations
  • Investigate specific examples of over-damped systems in mechanical or electrical contexts
USEFUL FOR

Students and professionals in physics and engineering, particularly those studying dynamics and control systems, will benefit from this discussion on over-damped differential equations.

swimforever
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λ=-(1/2τ)±√[(1/4∏^2)-(ω^2)]
I'm supposed to write the solutions of the differential equation for the over-damped case. The overdamped case is where ω<(1/2τ). I don't know how to write the solution.
I know that when ω=(1/2τ) we get the stuff under the square root to equal zero, but I am unsure of what happens or what the solution is when it's less than that.
 
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It would be helpful if you would show the differential equation to us.
 

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