Solving the differential equations involving SHM

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SUMMARY

The discussion focuses on the methodology of guessing solutions to differential equations in the context of damped and driven Simple Harmonic Motion (SHM). Participants emphasize the importance of prior experience with non-damped systems to make educated guesses about potential solutions. The sine wave is identified as the fundamental solution for SHM due to its repeating and continuous nature, while acknowledging that other waveforms like square and sawtooth waves are also valid in more complex scenarios. This highlights the iterative process of guessing and checking in mathematical problem-solving.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with Simple Harmonic Motion (SHM)
  • Knowledge of wave functions, particularly sine waves
  • Experience with mathematical problem-solving techniques
NEXT STEPS
  • Explore advanced techniques for solving differential equations in SHM
  • Study the characteristics and applications of various waveforms, including square and sawtooth waves
  • Learn about the role of damping in SHM and its mathematical implications
  • Investigate the use of Fourier series in representing complex waveforms
USEFUL FOR

Students of mathematics, physics enthusiasts, and educators seeking to deepen their understanding of differential equations and wave mechanics in the context of Simple Harmonic Motion.

anirocks11
What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?
 
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It is easy to check if a solution works once you make a guess. So generally you just have experience with non damped systems, and you guess and check until you get the form that works.
 
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anirocks11 said:
guessing certain solutions
That is a real problem when dealing with 'the next step' in Maths, for all students. "Why did they choose to do it that way?" we ask. Finding the way to the next step is always based on past experience so a student can't expect to find it to be very obvious.
In the SHM case, 'they' looked amongst the simplest function that would fit the rules - i.e. repeating and continuous etc etc. A single frequency sine wave is the first thing that would come to mind for SHM and it happens to be the right solution. When trying to solve the equations for Waves, there are many more possibilities (square waves, sawtooth etc) and they also work but we tend to start with a sine wave again - but we have to remember it's not the only one in that case.
 

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