# Concept derivation guides

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Concept guide



## [Linear harmonic oscillator: eligibility, k/m, residuals of a polynomial candidate and single-state energy](/en/derivations/derivation-mko-harmonic-oscillator-model-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-harmonic-oscillator-model)

Take m=2 and k=8. Check that the parameters satisfy the model's requirements and compute k/m; for the polynomial candidate x_c(t)=1−2t², compute the residual R(t)=m x_c''+k x_c, evaluate it at the specified points t=0, 1/2, 1, and prove that x_c is not a solution on any interval of positive length; prove that the equation 2x''+8x=0 has no nonzero polynomial solution; finally compute the energy E=4 of the single state (x,v)=(1,0), and use the energy-derivative identity to explain why a single energy value proves no conservation.

Linear harmonic-oscillator model

derivation-mko-harmonic-oscillator-model-01





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