Differential equations of harmonic oscillations

4 DYNAMICS OF THE HARMONIC OSCILLATION 2.4.1 Differential equations of harmonic oscillations 

It was shown earlier (refer to eq. (2.2.6)) that there exists a simple correlation between displacement and acceleration of an oscillating MP  

This equation is referred to as the differential equation of harmonic oscillations. By analyzing this equation, we can arrive at an important conclusion when solving a problem and arriving at an equation like that presented above, it means that the problem can be reduced to harmonic oscillations and the coefficient before the displacement function is the square of its cyclic frequency. 

This is proved by substitution of any of the proposed solutions into eq. (2.4.1). 

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Compare the expression obtained with the general type of differential equation of harmonic oscillation (eq. (2.4.1)). From the fact that both equations have a similar form, it can be stated that the weight makes a harmonic oscillation. Thereof, the other definition of a harmonic oscillation is an oscillation that occurs under the action of an elastic force. By equating the multipliers in the similar terms of the equation, we can derive an expression for the cyclic frequency of the spring pendulum ... 

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