MATH 21260: INTRODUCTION TO DIFFERENTIAL Recitation 2 Example. - CMU

MATH 21260: INTRODUCTION TO DIFFERENTIAL EQUATIONS

Recitation 2 Example. Consider the first order ordinary dierential equation

dy = y.

dt This is an autonomous ordinary dierential equation as the vector field f (t, y) = y does not depend on time. On the other hand, the ordinary dierential equation

dy = ty

dt

is

, as the vector field ( ) = depends on the independent

not autonomous

f t, y ty

variable . t

Direction field for =

Figure 1.

yy

Figure 2. Direction field for y = ty

Notice the following:

(1) In Figure 1 the vector field is constant along each horizontal line, whereas that is not the case in Figure 2 (why?).

(2) The vector field f (t, y) = y is equal to 0 when y = 0. When the vector field is equal to 0 the ODE reads y = 0, and y = constant solves this ODE trivially. We see this in Figure 1 as the solution curve along y = 0 is simply a horizontal line.

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MATH 21260: INTRODUCTION TO DIFFERENTIAL EQUATIONS

In light of the example above we make the following definitions.

Definition (Critical Point). A point ( ) is said to be a

of the vector

x, y

critical point

field if ( ) = 0. Notice that at a critical point we have that = constant

f f x, y

y

trivially solves the ODE above since the derivative of any constant is 0.

Next we examine the notion of stability. Consider the following ODEs and their associated vector fields:

dy = y, dy = y. dt dt

Direction field for =

Figure 3.

yy

In Figure 3, solution curves starting at a point close to the critical point = 0 y

move away from the critical point as ! 1. Here we say that the critical point is t

. unstable

MATH 21260: INTRODUCTION TO DIFFERENTIAL EQUATIONS

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Figure 4. Direction field for y = y

In Figure 4, solution curves starting at a point close to the critical point = 0 y

(from both sides) move towards from the critical point as ! 1. Here we say that t

the critical point is

.

asymptotically stable

If this is true only for points close to the critical point on one side then we say

that the critical point is

. (What's an example of an ODE that has a

semi-stable

semi-stable critical point?)

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MATH 21260: INTRODUCTION TO DIFFERENTIAL EQUATIONS

Exercise (Textbook 2.1 #21). Consider the first order ODE given by

dy = 2 3 y y.

dx (1) Find the critical points of the vector field.

(2) Draw a sign chart for the vector field.

(3) Sketch the one-dimensional phase portrait

(4) Sketch typical solution curves in the regions in the -plane determine by xy

the graphs of the equilibrium solutions.

(5) Classify each critical point as

,

, or

.

asymptotically stable unstable semi-stable

(1) 2 3 = 0 =) = 0 3. These are the critical points.

Solution.

yy

y,

(2) Sign chart: 2 3 = ( 3) 0 when 3 or 0, negative when

y y yy >

y> y<

0

3.

=)

y

ye

9 0 =) ( y

y>

ye

9) 0 >

=)

y

ln 9 >

or y < 0.

y e9 ye y

y

< 0 when 0 < y < ln 9.

e

(3) Sketch of phase portrait:

(4) Sketch:

Direction field for = ( y 9 ) y

Figure 6.

y ye y /e

(5) = 0 is asymptotically stable, = ln 9 is unstable (why?).

y

y

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