![\begin{displaymath}
{d^2y\over dx^2} - 2x{dy\over dx} + \lambda y = 0.
\end{displaymath}](h_1327.gif) |
(1) |
This differential equation has an irregular singularity at
. It can be solved using the series method
![\begin{displaymath}
\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n - \sum_{n=1}^\infty 2na_nx^n + \sum_{n=0}^\infty \lambda a_nx^n = 0
\end{displaymath}](h_1328.gif) |
(2) |
![\begin{displaymath}
(2a_2+\lambda a_4) + \sum_{n=1}^\infty [(n+2)(n+1)a_{n+2}-2na_n+\lambda a_n]x^n = 0.
\end{displaymath}](h_1329.gif) |
(3) |
Therefore,
![\begin{displaymath}
a_2 = - {\lambda a_0\over 2}
\end{displaymath}](h_1330.gif) |
(4) |
and
![\begin{displaymath}
a_{n+2} = {2n-\lambda\over (n+2)(n+1)} a_n
\end{displaymath}](h_1331.gif) |
(5) |
for
, 2, .... Since (4) is just a special case of (5),
![\begin{displaymath}
a_{n+2} = {2n-\lambda\over (n+2)(n+1)} a_n
\end{displaymath}](h_1331.gif) |
(6) |
for
, 1, .... The linearly independent solutions are then
If
, 4, 8, ..., then
terminates with the Power
, and
(normalized so that
the Coefficient of
is
) is the regular solution to the equation, known as the Hermite Polynomial. If
, 6, 10, ..., then
terminates with the Power
, and
(normalized so
that the Coefficient of
is
) is the regular solution to the equation, known as the Hermite Polynomial.
If
, then Hermite's differential equation becomes
![\begin{displaymath}
y''-2xy'=0,
\end{displaymath}](h_1345.gif) |
(9) |
which is of the form
and so has solution
© 1996-9 Eric W. Weisstein
1999-05-25