HP 50g Graphing Calculator - Manual (Page 534)

Page 16-57
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 * 2 2
0
= − = = −
n e
dx
d
e x H H
x
n
n
x n
n