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L
2
(x) = 1-2x+ 0.5x
2
L
3
(x) = 1-3x+1.5x
2
-0.16666…x
3
.
Weber’s equation and Hermite polynomials
Weber’s equation is defined as d
2
y/dx
2
+(n+1/2-x
2
/4)y = 0, for n = 0, 1, 2,
… A particular solution of this equation is given by the function , y(x) =
exp(-x
2
/4)H
*
(x/√2), where the function H
*
(x) is the Hermite polynomial:
In the calculator, the function HERMITE, available through the menu
ARITHMETIC/POLYNOMIAL. Function HERMITE takes as argument an integer
number, n, and returns the Hermite polynomial of n-th degree. For example, the
first four Hermite polynomials are obtained by using:
0 HERMITE, result: 1, i.e., H
0
*
= 1.
1 HERMITE, result: ’2*X’, i.e., H
1
*
= 2x.
2 HERMITE, result: ’4*X^2-2’, i.e., H
2
*
= 4x
2
-2.
3 HERMITE, result: ’8*X^3-12*X’, i.e., H
3
*
= 8x
3
-12x.
Numerical and graphical solutions to ODEs
Differential equations that cannot be solved analytically can be solved
numerically or graphically as illustrated below.
Numerical solution of first-order ODE
Through the use of the numerical solver (‚Ï), you can access an input
form that lets you solve first-order, linear ordinary differential equations. The
use of this feature is presented using the following example. The method used
in the solution is a fourth-order Runge-Kutta algorithm preprogrammed in the
calculator.
Example 1
-- Suppose we want to solve the differential equation, dv/dt = -1.5
v
1/2
, with v = 4 at t = 0. We are asked to find v for t = 2.
,..2,1),()1()(*,1*
22
0
=−==
−
ne
dx
d
exHH
x
n
n
xn
n