\( y ^ { \prime \prime \prime } - 3 y ^ { \prime \prime } + 3 y ^ { \prime } - y = 0 \)
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This third-order linear differential equation can be analyzed using characteristic equations. The characteristic equation derived from the given differential equation is \( r^3 - 3r^2 + 3r - 1 = 0 \), which factors into \( (r-1)^3 = 0 \). This indicates that \( r = 1 \) is a root of multiplicity three, leading to a general solution of the form \( y(t) = C_1 e^t + C_2 t e^t + C_3 t^2 e^t \), where \( C_1, C_2, \) and \( C_3 \) are constants determined by initial conditions. In practical applications, such differential equations often model systems in physics and engineering, such as damped oscillations or control systems. For example, when finding the response of a mechanical system to external forces, the above equation can represent how the position of a mass varies over time. Understanding the nature of the solutions helps engineers predict system behavior, ensuring stability or desired performance in real-world scenarios!
