Attainment of Steady State Algebraically: How Constant Coefficients in First and Second Order Differential Equations Reveal the Time Constant

Textbooks for differential equations often address transient and steady state solutions, but less frequently link the two or connect them to real world situations or laboratory results.In this paper, both types of solutions are considered, as well as the impact of the transient solution upon the full solution.This approach, as delineated, removes all of the complicated mathematics (which can now be accomplished using a calculator or computational software) in order to understand the general results, only requiring that the coefficients in the general equation be constant.The time constant, τ, is regularly taught in first order differential equations, and its relationship to the equation coefficients is widely known.However, for second order differential equations, the concept of the time constant appears to be rarely, if ever, discussed in many textbooks.Toward that end, in this work, second order differential equations with constant coefficients are examined.In addition, a typical equation for many real-world electrical and mechanical problems will be addressed by algebraically solving for τ.The steady state solution will be described in terms of the system's behavior after the transient no longer exists, i.e., after t = 5τ.This method avoids much mathematical manipulation, and has the added benefit of providing the solution for the steady state behavior of the system (t > 5τ).Furthermore, it reveals the parameters needed to be adjusted in order to alter the time constant by observing its algebraic expression and how the adjustment affects other terms.The importance of the time constant will be emphasized for both first and second order differential equations, and will demonstrate its utility in real-world systems, using several electrical and mechanical examples.