Given a Second-Order Ordinary Differential Equation
![\begin{displaymath}
{\tilde {\mathcal L}} u(x) \equiv p_0 {du^2\over dx^2} +p_1 {du \over dx} +p_2 u,
\end{displaymath}](a_382.gif) |
(1) |
where
and
, the adjoint operator
is defined by
Write the two Linearly Independent solutions as
and
. Then the adjoint
operator can also be written
![\begin{displaymath}
{\tilde {\mathcal L}}^\dagger u = \int (y_2 {\tilde {\mathca...
...\,dx
= {\left[{{p_1\over p_0} ({y_1}'y_2-y_1{y_2}')}\right]}.
\end{displaymath}](a_389.gif) |
(3) |
See also Self-Adjoint Operator, Sturm-Liouville Theory
© 1996-9 Eric W. Weisstein
1999-05-25