# Laguerre's Equation - HP 50g User Manual

Graphing calculator.

## Laguerre's equation

Laguerre's equation is the second-order, linear ODE of the form x (d
+(1 x) (dy/dx) + n y = 0. Laguerre polynomials, defined as
L
0
are solutions to Laguerre's equation. Laguerre's polynomials can also be
calculated with:
The term
is the m-th coefficient of the binomial expansion (x+y)
number of combinations of n elements taken m at a time. This function is
Chapter 17).
You can define the following function to calculate Laguerre's polynomials:
When done typing it in the equation writer press use function DEFINE to create
the function L(x,n) into variable @@@L@@@ .
To generate the first four Laguerre polynomials use, L(x,0), L(x,1), L(x,2), L(x,3).
The results are:
(
)
, 1
(
x
L
x
n
L
(
x
)
n
n
(
n
1
n
x
4
n
⎜ ⎜
⎟ ⎟
m
m
L
(x) = .
0
L
(x) = 1-x.
1
x
n
n
(
e
d
x
)
!
n
dx
n
m
n
(
) 1
⎜ ⎜
m
m
!
m
0
) 1
2
x
...
n
!
C
( !
n
m
)!
x
)
e
,
1
2 ,
,...
n
n
m
⎟ ⎟
x
.
n
(
) 1
n
....
x
n
!
(
n
,
m
)
n
. It also represents the
2
2
y/dx
)
,
Page 16-56