\( y ^ { ( 4 ) } - 4 y ^ { \prime \prime \prime } + 4 y ^ { \prime \prime } = 0 \)

This equation is a linear homogeneous fourth-order ordinary differential equation, frequently encountered in physics and engineering contexts. The general solution can involve combinations of polynomial and exponential functions, which arise from related characteristic equations. Such equations can model systems with damping, oscillations, or other behaviors seen in systems like beams and circuits. When tackling this ODE, a common mistake is to not check the roots for repeated factors after finding the characteristic polynomial. Remember, if you find a root with multiplicity, it requires special handling by adding polynomial terms to your solutions to account for that multiplicity. Additionally, make sure to simplify the equation properly so you can accurately apply the appropriate methods, whether that’s the method of undetermined coefficients or variation of parameters!