equilibrium solutions and stability example 2.1. newton’s
TRANSCRIPT
2. Equilibrium Solutions and Stability
Example 2.1. Newton’s law of cooling
dx
dt= �k(x� A), (k > 0 constant)
x(0) = x0
By separation of variables, we have the explicit solution
x(t) = A+ (x0 � A)e�kt
It follows that
limt!1
x(t) = A
(a) Typical solution curve.
(b) Phase diagram.
1
,
,
•
→environmental temperature ( coast)
*¥÷÷¥:* - fof anobject
¥b= At
+→ o ,-ht → -o, e-
"→ o
Definition 2.1. Any first-order ODE which can be written:
(2.1)dx
dt= f(x),
where the independent variable t does not appear explicitly, is called autonomous.
Any autonomous ODE is automatically separable, with the (implicitly defined)
general solution:
(2.2)
Z1
f(x)dx = t+ c.
Definition 2.2. The solutions of the equation f(x) = 0 are called critical pointsof the autonomous di↵erential equation (2.1).
Definition 2.3. If x = c is a critical point of Eq. (2.1), then the di↵erential
equation has the constant solution x(t) = c. A constant solution of a di↵erential
equation is sometimes called an equilibrium solution.
Example 2.2.
(2.3)dx
dt= kx (M � x)
Find the critical points.
2
=( separable but not
autonomous) dxIt
=✗'
→ Separable equation
f- is not dependingont
.
independent
± ☐pits of the ODE -
= ( Po - Ax - Vol , p
- =f
use the definition to find thethe
critical points
are actually theconstant
critical pts .solutions of the ODE .
some fi check,
b. ✗ ( in-14--0 ✗12=1"
⇒ {✗1=0
→" ""
uts -_d¥= 0→ii.Hiiiii pints
✗2 = M of the ODERHS = b. ii. in,
"
= 0
⇒ × ,= inis asolati
.
of the ODE.
Definition 2.4. The critical point c is stable if, for each ✏ > 0, there exists � > 0
such that
|x0 � c|< � implies that |x(t)� c|< ✏
for all t > 0. The critical point x = c is unstable if it is not stable.
Example 2.3. Determine the stability of critical points in Example 2.2.
3
if there is an errorin the initial condition,
but the solution will go to thescene value
¥.-0 -
•Cic initial
y,wud:t:n of
Hut"" ")glue line :
solution✗
✓ of +↳ efuat""
12
c.is,€¥É- > > > > as time ct, → 0
g.adit""
1in 1 , =- Iim 12=/im ↳
÷
T it:me)
Harvesting a Logistic Population:
The autonomous di↵erential equation
(2.4)dx
dt= ax� bx2 � h
(with a, b, and h all positive) may be considered to describe a logistic population
with harvesting. It can be re-written in the form
(2.5)dx
dt= kx (M � x)� h
Example 2.4. Solve the di↵erential equation (2.5). Find the critical points of
(2.5).
4
~
= fix) →auto .
= 1i±_Fykcritical points .
N= MÉ4h/kkxlm
-x) -hiopts
= %fµg-}'cat:c,
=
- hx' -1km
-X- h :O
=
_kr±kFÉ¥_1° <
H.ee am
HIN fix = KHS of 4)É
-2k dx = him -4.1×-1-1 )
✗solution It
✗CH Y %. stable :
01¥÷•÷És>↳ > s > > •
no matterwhat 7-'C' is
the solut!- will finally
↳*
convey tothe witted
to
point .
whenF-°
A
⇒µµµ,⇒*N is stable
.
②✗e NI
µ-✗ so,
X-H >°
¥ = klN-×t)
⇒pits
> 0⇒
so
when f- 0 ,
⇒ ✗is}
① ✗ > Nj"*kHs, w-✗co, X-H"
e-
⇒ pasco ,⇒ ¥+10 -7×4
Harvesting a Logistic Population:
The autonomous di↵erential equation
(2.4)dx
dt= ax� bx2 � h
(with a, b, and h all positive) may be considered to describe a logistic population
with harvesting. It can be re-written in the form
(2.5)dx
dt= kx (M � x)� h
Example 2.4. Solve the di↵erential equation (2.5). Find the critical points of
(2.5).
4
1-1--4×1① critical points .
solutions of f-1×1=0
② critical points → equilibrium
solution of the ODE .
③ state
coast k>0
✗ o- H > 0
DX
It=kTÉH -1×-1--1)
µ-X o 20
µ ,H are
2Cbittcul points
⇒ RHS > 0
DX
µ ,-N >
H ⇒j+
we start from a pointstudy ✗
= H
✗o > ✗=H, however
the
①xd = Xo
< H
functionisincreasing
✗ o- H
< °
⇒ ×=Hisuotstaµ-to
>° It co
⇒ pitsco⇒ It
÷÷¥ > >''¥
⇒✗ I
⇒ *µ is
not state' ×÷ci >so If
÷
④ >✗a)=×
Example 2.5. The di↵erential equation
(2.6)dx
dt= x (4� x)� h
(with x in hundreds) models the harvesting of a logistic population with k = 1 and
limiting population M = 4 (hundred). For di↵erent harvesting level h, what arethe equilibrium solutions and critical points?
5
Éh=o ① h > 4 .
4 -h - o ② when
=) no real solution . ✗(of =XoC2
✗'-4×-14--0
② h < 4 , N>4 ⇒ base
-1×-25
Hiv = ÉÉ <0
③ h= 4⇒ ¥+0
=z t Fh ⇒N=H=z-
⇒ ✗ toµ : 2 -1 4Th ① ✗c) = Xo > 2 .
g. ⇒ µ, , ,,,, , ⇒ ×,
,
notstable
.
¥: mi⇒ ¥+0 ⇒ ✗ o
v
+"' "
. > > . > >✗ =L
•
xd N,
÷