91Ó°ÊÓ

The first step in solving the given differential equation, \( y^{\text{′′′}} + y^{\text{′′}} + 4 y^{\text{′}} + 4 y = 0 \), is transforming it into its characteristic equation. This is a polynomial equation that helps in finding the nature of the solutions to the differential equation. By assuming a solution of the form \( y = e^{rt} \), when we differentiate and substitute back into the original differential equation, we obtain the characteristic polynomial: \[ r^3 + r^2 + 4r + 4 = 0 \]Understanding the characteristic equation is crucial as it gives us the necessary polynomial whose roots determine the behavior and structure of the solutions to the differential equation. It converts a differential problem into an algebraic one, streamlining the pathway to finding the solution.

The next essential step is finding the roots of the characteristic equation, \[ r^3 + r^2 + 4r + 4 = 0 \]. Solving this cubic polynomial is paramount as the roots indicate the form of the solutions to our differential equation. To solve it, methods like factoring or the Rational Root Theorem can be used. In this specific example, by factoring, we get: \[ (r+2)(r^2-1) = 0 \]Which simplifies to: \[ r + 2 = 0 \] and \[ r^2 - 1 = 0 \], leading us to the roots: \[ r = -2, \ r = 1, \ r = -1 \]The nature of these roots (whether real, repeated, or complex) directly influences the form of the general solution to the differential equation.

Having obtained the roots of the characteristic equation \[ r = -2, \ r = 1, \ r = -1 \], we can now construct the general solution of the differential equation. The general solution is a linear combination of the exponentials formed by these roots. Hence, for our roots, the general solution is: \[ y(t) = C_1 e^{-2t} + C_2 e^{t} + C_3 e^{-t} \]Here, \( C_1, C_2, \) and \( C_3 \) are arbitrary constants that can be determined if initial conditions are provided. This combination of exponentials reflects the influence of each root on the system's behavior, and by finding the particular values of the constants, we can tailor this general solution to fit specific initial conditions.