# Math Help - ODEs, Adjoint forms and matrix representation

1. ## ODEs, Adjoint forms and matrix representation

Hi MHF members,

I was looking up in the books I have but I could not go anywhere.
Consider the higher-order ode
$L[y]:=y^{(n)}+\sum_{k=0}^{n-1}p_{n-k}y^{(k)}=0$,
which has the matrix form
$\pmb{Y}^{\prime}+\pmb{A}\pmb{Y}=\pmb{0}$,
where
$\pmb{Y}=\left( \begin{array}{c} y \\ y^{\prime} \\ \vdots \\ y^{(n-1)} \\ \end{array} \right)$ and $\pmb{A}=\left( \begin{array}{cccc} & -1 & & \\ & & \ddots & \\ & & & -1 \\ p_{n} & p_{n-1} & \cdots & p_{1} \\ \end{array} \right)$.
The adjoint form of $L$ is given by
$L^{\ast}[z]=z^{(n)}+\sum_{k=0}^{n-1}(-1)^{n-k}[p_{n-k}z]^{(k)}=0$,
where the coefficients are assumed to be real and sufficiently smooth.
How the matrix form of the adjoint equation can be derived?
Have any of you met with this in a book/reference?

Thanks.
bkarpuz

2. ## Re: ODEs, Adjoint forms and matrix representation

Originally Posted by bkarpuz
Hi MHF
How the matrix form of the adjoint equation can be derived?
Actually, I can use the Leibnitz rule for higher-order derivatives of
the product term $p_{n-k}z$ for expansion, and then reverse the order of two sums and get
$L^{\ast}[z]=z^{(n)}+\sum_{k=0}^{n-1}\underbrace{\bigg[\sum_{\ell=k}^{n-1}(-1)^{n-\ell}\binom{\ell}{k}p_{n-\ell}^{(\ell-k)}\bigg]}_{\text{Coefficient}}z^{(k)}$.
Then write the matrix form of this equation. However, I was looking for a different way.
For instance, having the coloumns of the unknown both involving $p$ and $z$, if possible.