1. ## Differential Equation

$y^{(n)} + p_{n-1}(x)\,y^{(n-1)} + \cdots + p_1(x)\,y' + p_0(x)\,y = 0,$ (1) be the n-th order homogeneous differential equation which is represented in matrix form $\dot{\vec{y}} = A \vec{y}$ (2)

If $y_1,...,y_n$ are the solutions to (1), what are the correspondence solutions of (2).

I know if $y$ is solution to (1) then the corresponding solution $\vec{y}$for (2) is given the vector (y,y',y'',y''',......y^n-1)^T

2. Originally Posted by charikaar
$y^{(n)} + p_{n-1}(x)\,y^{(n-1)} + \cdots + p_1(x)\,y' + p_0(x)\,y = 0,$ (1) be the n-th order homogeneous differential equation which is represented in matrix form $\dot{\vec{y}} = A \vec{y}$ (2)

If $y_1,...,y_n$ are the solutions to (1), what are the correspondence solutions of (2).

I know if $y$ is solution to (1) then the corresponding solution $\vec{y}$for (2) is given the vector (y,y',y'',y''',......y^n-1)^T

??? If $y_1,...,y_n$ are the solutions to (1), then the corresponding solutions to 2 are the vectors $(y_i, y'_i, y"_i, ..., y^{n-1}_i)^T$ for all i, of course. That can be written as a matrix with those vectors as columns.
$\left(\begin{array}{ccc}y_1&\cdots&y_n\\y_1\prime& \cdots&y_n\prime\\\vdots&\vdots&\vdots\\y_1^{(n-1)}&\cdots&y_n^{(n-1)}\end{array}\right)$