Can a 2nd Order Linear ODE Be Expressed in Terms of σ and ω?

[Jhenrique]

If a 2nd order linear ODE:

can written in terms of natural frequency ω₀ and damping ratio ζ:

being:

So, it too can be written in terms of exponential decay/growth σ and angular frequency ω?

 

[jvicens]

I can confirm that a 2nd order linear ODE can indeed be written in terms of natural frequency ω0 and damping ratio ζ. This is a common representation used in the study of oscillatory systems, such as mass-spring systems or electrical circuits. In this representation, the natural frequency ω0 represents the frequency at which the system would oscillate in the absence of any external forces, while the damping ratio ζ represents the amount of energy lost in each oscillation due to damping.

Additionally, it is also possible to write the 2nd order linear ODE in terms of exponential decay/growth σ and angular frequency ω. This representation is commonly used in the study of exponential functions and their applications in various fields, such as population growth or decay, radioactive decay, and financial growth or decay.

In this representation, the exponential decay/growth σ represents the rate at which the system decays or grows exponentially, while the angular frequency ω represents the frequency of oscillation in the exponential function. This representation is useful in understanding the behavior of systems that exhibit exponential growth or decay, and can also be used to solve for various parameters in the system.

In summary, both representations of the 2nd order linear ODE have their own advantages and uses in different fields of study. it is important to be familiar with both representations and understand how they can be applied to different systems and problems.