194 ■ Slope fields the Easy Way on the
TI-89 and StudyCard™ Sharing
60-Minute Hands-On ■ IntermediateTI-89 ■ Calculus, Physics
Matching slope fields handout with accompanying TI-89 script
will make graphing and teaching about differential equations
for the AP Calculus exam so much easier.
StudyCards™ for AP Physics and Calculus will also be shared.
Slope Fields: Motivation, Activity,
APcalcAB04#6, DE matching, TI89
“Geometric interpretation of differential
equations via slope fields and the
relationship between slope fields and
solutions curves for differential
equations.”
This topic has been part of the topic
outline for Calc BC since the 1998 exam
Sean Bird
AP Physics & Calculus
Covenant Christian High School
7525 West 21st Street
Indianapolis, IN 46214
Phone: 317/390.0202 x104
Email: seanbird@covenantchristian.org
Website: http://covenantchristian.org/bird
Psalm 111:2
Three/Four Pronged approach



We now know how to algebraically solve
separable differential equations.
Today we will graphically find the particular
solution for any DE using SLOPE FIELDS.
In calcBC we will then numerically solve DEs
using Euler’s Method
Verbally
(Recall new sign chart policy)
Slope Fields
• If you enjoyed connecting the dots,
you’ll love slope fields
• It is a graphical method to find a
particular solution to any differential equation.
• a DE is nothing but a slope equation dy
y
 lim
dx x 0 x
Given:
dy
2
 x ( y  1)
dx
Let’s sketch the slope field …
But how?


Substitute the x and y into the
differential equation for each
of the points.
Plot this slope on the graph.
dy
2
 x ( y  1)
dx
dy
dx
 x 2 ( y  1)
dy
dx
dy
dx
x 1
y 3
x 1
y 2
dy
dx
 (1) 2 (3  1)  2
 (1) 2 (2  1)  1
x 1
y 1
 ( 1) 2 (1  1)  0
dy
dx
dy
dx
dy
dx
x 1
y 0
x 0
y 0
 (1) 2 (0  1)  1
 (0) 2 (0  1)  0
x 1
y 0
 (1) 2 (0  1)  1
dy
dx
 x 2 ( y  1)
When are the slopes positive?
i)
1st the slopes will be positive
when the differential equation
is positive, i.e.
x 2 ( y  1)  0
ii) And x2 is always positive
(except when x=0)
iii)
y – 1 > 0 when y > 1
ANSWER:
y > 1, but x ≠ 1
dy
 x 2 ( y  1)
dx
Given f(0)=3, what is the particular solution.
Separate the variables
dy
 x 2dx
y 1
1
2
dy

x
 y  1  dx
1 3
ln y  1  x  C
3
1 x3 C
1 x3
3
Exponentiate both sides y  1  e
 e 3 eC
1 x3
C
 y  1  Ke 3 , apply IC f (0)  3
Let K  e

y  2e
1 x3
3
1
E
C
The thought
process…
A
G
i)
B
Isoclines along horizon. lines
→ DE depends only on y
D
ii) Same slope along vert. line
→ DE depends only on x
I
iii) Sinusoidal
H
iv) Consider a specific point –
what is the slope there.
F
K
J
v) What makes the slope zero?
What values of x and y make
the slope 1?
vi) Note y'(x) = x+y has same
slopes along the diagonal.
vii) Solve the separable DE
Running the DEmatch script
1. Select TEXT EDITOR under APPS
2. Open…
3. Folder: calculus, Variable: dematch
4. Green (chartreuse for Titanium)
diamond, Up arrow will quickly jump
cursor to the top. 2nd, up arrow scrolls up a page.
5. Press F4 all the way down
6. Note what each step accomplishes
and READ important rules at the end.
StudyCard Stacks
Algebra:
slope,
slope/intercept for 89
for 83/84 new!
Trig/PreCalc:
study of functions
Calculus:
identity **
limits83 **
limits89 **
derivative **
integral **
Physics:
AP Physics for the TI89 ** 83*
Prefix for the 83/84 **
History of Physics for the 89 **
or 83/84 **new!