Hermite Differential Equation
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
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12-6-2018
2071
Hermite Differential Equation
The second-order ordinary differential equation
 |
(1)
|
This differential equation has an irregular singularity at 
. It can be solved using the series method
 |
(2)
|
![(2a_2+lambdaa_0)+sum_(n=1)^infty[(n+2)(n+1)a_(n+2)-2na_n+lambdaa_n]x^n=0.](http://mathworld.wolfram.com/images/equations/HermiteDifferentialEquation/NumberedEquation3.gif) |
(3)
|
Therefore,
 |
(4)
|
and
 |
(5)
|
for 
, 2, .... Since (4) is just a special case of (5),
 |
(6)
|
for 
, 1, ....
The linearly independent solutions are then
These can be done in closed form as
where 
is a confluent hypergeometric function of the first kind and 
is a Hermite polynomial. In particular, for 
, 2, 4, ..., the solutions can be written
where 
is the erfi function.
If 
, then Hermite's differential equation becomes
 |
(14)
|
which is of the form 
and so has solution
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