Direction Fields
Studying Solutions Without Finding Them (Explicitly)
First Things First…
What’s A Differential
Equation?
Any equation that involves a
derivative is a differential equation
Derivatives Tell Us Change
Rate of
Change
of y
Current
time
Value of y
(or other
independent
variable)
Derivatives Tell Us Change
Rate of
Change
of y
Current
time
Value of y
(or other
independent
variable)
Secret: We don’t know
how to solve many
ODEs!
Direction Fields
Suppose
(time) t = 0.5,
(value) y = 2.1
y
2.1
Then
ç
0.5
t
Direction Fields
Suppose
(time) t = 0.5,
(value) y = 2.1
y
2.1
Then
ç
0.5
t
Direction Fields
Question: If
y
2.1
ç
estimate
0.5
t
Direction Fields
Question: If
y
2.1
estimate
ç
Change = Rate x Time = 1.0 x 1.05 = 1.05
0.5
t
Direction Fields
Question: If
y
2.1
estimate
ç
So at t = 1.5, Value = 2.1 + 1.05 = 3.15
0.5
t
Direction Fields
Question: If
y
2.1
estimate
ç
Tangent Line:
0.5
t
Direction Fields
Question: If
3.15
y
2.1
ç
estimate
ç
0.5
1.5
t
Direction Fields
3.15
y
2.1
ç
ç
0.5
“Direction”
the system is moving.
1.5
t
Direction Fields
y
2.1
ç
0.5
Slope is
when
t
Direction Fields
y
2.1
ç
0.5
Slope is
when
t
Direction Fields
Can find
directions at
many different
points.
y
2.1
ç
0.5
t
Direction Fields
Can find
directions at
many different
points.
y
2.1
ç
ç
y=2.1, t=1, so y’ = 0
0.5
1.0
t
Direction Fields
Can find
directions at
many different
points.
y
2.1
ç
ç
ç
y=2.1, t=1.5,
so y’ = -1.05
0.5
1.0
1.5 t
Direction Fields
y=3.1, t=0.5,
so y’ = 1.55
3.1
y
ç
2.1
ç
ç
0.5
1.0
ç
1.5 t
Direction Fields
4.1
ç
3.1
y
ç
2.1
ç
ç
0.5
1.0
y=4.1, t=0.5,
so y’ = 2.05
ç
1.5 t
Direction Fields
4.1
ç
ç
ç
3.1
y
ç
ç
ç
2.1
ç
ç
ç
0.5
1.0
1.5 t
Filling out the
grid gives a
“Direction Field”
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
This is a full
Direction Field
t
Direction Fields
2.0
y
t = 0, y = 1.0
1.0
0.5 1.0 1.5 2.1
Following the
arrows shows
how system
behaves.
t
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
Following the
arrows shows
how system
behaves.
t
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
Following the
arrows shows
how system
behaves.
t
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
t
Note: Eventually,
y=0
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
t
y=0
y’ = (1-t)*0
y’=0
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
t
If y’=0,
y Stops
Changing
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
Thus, y values that make
y’ = 0 are called
Equilibrium Values
t
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
If the direction field is heading
towards the Equilibrium Value, the
Equilibrium
is stable
t
Direction Fields
2.0
y
1.0
0.5 1.0 1.5 2.1
The solution y(t) = Equilibrium Value
is called an Equilibrium Solution
t
Direction Fields
2.0
Here, y(t) = 0
is an Equilibrium
Solution
y
1.0
0.5 1.0 1.5 2.1
The solution y(t) = Equilibrium Value
is called an Equilibrium Solution
t
Direction Fields
2.0
Here, y(t) = 0
is an Equilibrium
Solution
y
1.0
0.5 1.0 1.5 2.1
The solution y(t) = Equilibrium Value
is called an Equilibrium Solution
t
So, Without Finding Solution
•
We can…
•
Draw Direction Fields
•
Find Equilibrium Values and Equilibrium Solutions
•
Determine If Equilibrium Values and Solutions are
Stable or Unstable
Questions?