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Math Help - Differential Equation

  1. #1
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    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

    Thank you for your help in advance.
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  2. #2
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    Quote Originally Posted by charikaar View Post
    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

    Thank you for your help in advance.
    ??? 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.
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  3. #3
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    Is this the correct matrix for solutions
    \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)

    many thanks
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