A homogeneous linear differential equation is a type of differential equation that can be expressed in the form $a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0$, where the function on the right-hand side is zero. These equations are important in understanding the behavior of linear systems and can be solved using various methods, including the eigenvalue approach, which provides a systematic way to find solutions through the characteristics of associated matrices.
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