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
Содержание 50G
Страница 1: ...HP g graphing calculator user s guide H Edition 1 HP part number F2229AA 90006 ...
Страница 130: ...Page 2 70 The CMDS CoMmanDS menu activated within the Equation Writer i e O L CMDS ...
Страница 177: ...Page 4 10 Function DROITE is found in the command catalog N Using EVAL ANS 1 simplifies the result to ...
Страница 206: ...Page 5 29 LIN LNCOLLECT POWEREXPAND SIMPLIFY ...
Страница 257: ...Page 7 20 ...
Страница 383: ...Page 11 56 Function KER Function MKISOM ...
Страница 715: ...Page 21 68 Whereas using RPL there is no problem when loading this program in algebraic mode ...
Страница 858: ...Page L 5 ...