Linear Stochastic Differential Equations and Linear Random Processes

We know that the general solution of the linear differential equation (12.1)$$ x^{\left( n \right)} \left( t \right) - a_1 \left( t \right)x^{\left( {n - 1} \right)} \left( t \right) - \ldots - a_n \left( t \right)x\left( t \right) = 0,\,t > t_0 $$ (with constant coefficients) can be written in the form (12.2)$$ x\left( t \right) = \sum\limits_{k = 0}^{n - 1} {\omega _k \left( {t,t_0 } \right)x_k ,\,t \geqslant t_0 ,} $$ where we denote by x₀, …, x_n-1 the initial values $$ x_0 = x\left( {t_0 } \right), \ldots ,x_{n - 1} = x^{\left( {n - 1} \right)} \left( {t_0 } \right) $$ and by ω_k(t,t₀) special solutions with initial value x_k = 1, x_j, = 0, for j ≠ k.