A The differential equation of Damped Harmonic Oscillator

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For the differential equation of a damped harmonic oscillator, the condition b^2/m > 4*k leads to a classic solution, while the opposite requires complex numbers. The validity of a solution can be confirmed by substituting it back into the original equation, similar to how differentiation and integration relate. The formulation of the differential equation is rooted in physics, particularly Hooke's law, which states that the force is proportional to the spring's stretch, supported by experimental verification. A relevant resource for further understanding is provided in a linked MIT course on differential equations. This discussion highlights the interplay between mathematical solutions and physical principles in modeling damped oscillations.
Gh. Soleimani
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If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?
 
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I'm not sure what you mean by proving it.

If you find a solution you can always insert it into the differential equation to see if its true in the same way that the integral of some function when differentiated will produce the same function.

If you mean how did the differential equation for a damped harmonic come to be defined that would be more physics related with assumptions for the force of the spring (Hooke's law ie force proportional to spring stretch) and the motion verified by experiment.
 
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