Understanding Simple Harmonic Motion: Solving the Second Order ODE

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SUMMARY

The equation of simple harmonic motion is represented by the second order ordinary differential equation (ODE) (d²x)/(dt²) + xk² = 0. The general solution can be derived by using an ansatz of the form x = Ae^(rt), where A is a constant and r is a parameter to be determined. Substituting this ansatz into the ODE allows for the identification of the constants based on initial conditions. This method provides a clear pathway to solving the equation effectively.

PREREQUISITES
  • Understanding of second order ordinary differential equations (ODEs)
  • Familiarity with the concept of simple harmonic motion
  • Knowledge of exponential functions and their properties
  • Ability to apply initial conditions to mathematical solutions
NEXT STEPS
  • Study the derivation of solutions for second order linear ODEs
  • Explore the applications of simple harmonic motion in physics
  • Learn about the role of initial conditions in determining unique solutions
  • Investigate numerical methods for solving ODEs when analytical solutions are complex
USEFUL FOR

Students of physics and mathematics, educators teaching differential equations, and engineers applying simple harmonic motion principles in their designs.

coverband
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The equation of simple harmonic motion is given as

(d2x)/(dt2)+xk^2=0

Could someone show me HOW we arrive at the general solution to this second order ODE
 
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coverband said:
The equation of simple harmonic motion is given as

(d2x)/(dt2)+xk^2=0

Could someone show me HOW we arrive at the general solution to this second order ODE
The solution is straightforward, simply take an Anzats of the form

[tex]x=Ae^{rt}[/tex]

and substitute into the ODE. The value of the constants may be determined using appropriate initial conditions.
 

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