Example Of A 2Nd-Order Equation - Texas Instruments TI-89 Manual Book

Example of a 2nd-Order Equation

Example
Note: t0 is the time at which
the initial conditions occur. It
is also the first t evaluated
for the graph. By default,
t0=0.
Important: For 2nd-order
equations, you must set
or
Fields= DIRFLD FLDOFF
Important: Fields=
DIRFLD
cannot plot a time axis. An
Invalid Axes error occurs if
Axes= or if t is set as a
TIME
axis.
CUSTOM
The 2nd-order differential equation y''+y = 0 represents a
simple harmonic oscillator. Transform this into a system of
equations for the Y= Editor. Then, graph the solution for initial
conditions y(0) = 0 and y'(0) = 1.
1. Press 3 and set
2. Define a system of equations
for the 2nd-order equation as
described on page 186.
Rewrite the equation and
make the necessary
substitutions.
3. In the Y= Editor ( ¥ # ),
enter the system of equations.
4. Enter the initial conditions:
and
yi1=0 yi2=1
5. Press:
ƒ
9
.
or
— —
¥ Í
TI-89:
¥
TI-92 Plus:
and set Axes = ON
,
OFF Solution Method = RK
=
Fields DIRFLD
6. In the Y= Editor, press:
: 2 ‰
TI-89
TI-92 Plus: ‰
and make sure
with and
y1 y2
7. In the Window Editor
( ¥ $ ), set the
Window variables.
8. Display the Graph screen
( ¥ % ).
If you select ZoomSqr
is actually a circle. However,
variables.
Graph=DIFF EQUATIONS
y'' + y = 0
y'' = ëy
y'' = ëy1
y2' = ëy1
F
, Labels =
, and
.
Axes = CUSTOM
as the axes.
t0=0.
tmax=10.
tstep=.1
tplot=0.
x axis = y1 = y
y axis = y2 = y'
( „
5 ), you can see that the phase-plane orbit
will change your Window
ZoomSqr
Chapter 11: Differential Equation Graphing
.
yi1 is the initial
condition for y(0).
yi2 is the initial
condition for y'(0).
xmin=ë 2. ncurves=0.
xmax=2. diftol=.001
xscl=1. fldres=14.
ymin=ë 2. dtime=0.
ymax=2.
yscl=1.
187