Mathe IIILecture 6Mathe IIILecture 6

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Example

dx= B x - a x - b

dt

1dx = Bdt

x - a x - b

1 1 1 1= -

x - a x - b b - a x - b x - a

1 1 1 1

dx = - dxx - a x - b b - a x - b x - a

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Example

dx= B x - a x - b

dt

1

1dx = Bdt C

x - a x - b

1 1 1 1

dx = - dxx - a x - b b - a x - b x - a

ln ln 1

x - b x - ab - a

ln1 x - b

b - a x - a

ln 1

1 x - b= Bt + C

b - a x - a

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Example

dx= B x - a x - b

dt

ln 1

1 x - b= Bt + C

b - a x - a

ln 2

x - b= B b - a t + C

x - a

2 B b-a tCx - b= e e

x - a

2 B b-a t B b-a tCx - b= e e Ce

x - a

2 1C = C b - a

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Example

dx= B x - a x - b

dt B b-a tx - b

= Cex - a

B b-a t

b - ax = a +

1 - Ce

assume B = -1, a = -1, b = 2

x = x + 1 2 - xiffx > 0 - 1 < x < 2

3 t

3x = -1+

1 - Ce

x = - x + 1 x - 2

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Example

x = x - 1 2 - x iffx > 0 - 1 < x < 2

3 t

3x = -1+

1 - CeC < 0

t

x

2

-1

+C > 0

(1/3)lnC

3

ln

t1 - Ce = 0

1t = C

3

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First Order Linear Differential Equations

x + a t x = b t

Assume that is a constant, a t a t a.

at atx + ax e = b t e

But at at

tx + ax e = xe

ateX x + ax = b t

hence at at

txe = b t e

at at= bx + a e t ex

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is an of the equationate Integrating Factor

x + ax = b t

at at

txe = b t e

at atxe b t e dt +C

-at atx t e b t e dt +C

x + ax = b t

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x + ax = b t

-at atx t e b t e dt +C

when b t b

-at atx t e b e dt +C -at atb

e ea

+C

-atbx t = e

a+C

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Price AdjustmentExample:

Demand : D p = a - bp

p = λ D p - S p

λ b+ β tex p = λ a - bp - α - βp

p + λ b + β p = λ a - α

λ b+ β t λ b+ β tpe = λ a - α e

λ b+ β t λ b+ β tpe = λ a - α e dt + C

Supply : S p = α + βp

iff p > 0 D p > S p

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Price AdjustmentExample:

λ b+ β t λ b+ β tpe = λ a - α e dt + C

λ b+ β t

λ b+ β t epe = λ a - α + C

λ b + β

-λ b+ β ta - αp t = + Ce

b + β

limt

a - αp t =

b + β

D p* = a - bp* = α + βp* = S p*

= p*

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x + a t x = b t a(t)dtex

a(t)dt a(t)dt

t

x + a t x e xe

a(t)dt a(t)dt

t

xe b t e

a(t)dt a(t)dt

t

xe b t e

a(t)dt a(t)dt

xe b t e dt + C

is a constant a nott

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x + a t x = b t a(t)dtex

a(t)dt a(t)dt

xe b t e dt + C

a(t)dt a(t)dt

x e b t e dt + C

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x + a t x = b t 0 0x t = x

a(t)dt a(t)dt

x e b t e dt + C

We found that:

let A t A t = a(t)dt F t = b t e dt

-A tx t = e F t + C

0-A t0 0x = e F t + C

0A t0 0C = x e - F t

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x + a t x = b t 0 0x t = x

-A tx t = e F t + C

0A t0 0C = x e - F t

0A t-A t0 0x t = e F t + x e - F t

0- A t - A t -A t

0 0x t = x e + e F t - F t

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0- A t - A t -A t

0 0x t = x e + e F t - F t

but A t F t = b t e dt

0

tA s

0

t

F t - F t b s e ds

0

t-A t -A t A s

0

t

e F t - F t e b s e ds

0

t- A t - A s

t

b s e ds

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0- A t - A t -A t

0 0x t = x e + e F t - F t

but A t F t = b t e dt

0

tA s

0

t

F t - F t b s e ds

0

t-A t -A t A s

0

t

e F t - F t e b s e ds

0

t- A t - A s

t

b s e ds

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0

0

t- A t - A t - A t - A s

0

t

x t = x e + b s e ds

but A t = a(t)dt

t

s

A t - A s = a(ξ)dξ

t t

t0 s

0

- a(ξ)dξ t - a(ξ)dξ

0

t

x t = x e + b s e ds

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Stability and Phase Diagrams

x = F x

is a stationary point (an equilibrium)

of the equation if

x = a

F a = 0

if and

then for all 0 x t = a F a = 0

t : x(

t)

a

is stable ??? aif does it move towards with time? x(t) = a + ε a

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Stability and Phase Diagrams

x = F x

x

y = x

F x

stable stationary point

unstable stationary point

x = F x > 0

x = F x < 0 F a < 0 F a > 0

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Stability and Phase Diagrams

x = F x

stable stationary point

unstable stationary point

F a = 0 F a < 0

F a = 0 F a > 0

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p = H D p - S p

ExampleGeneral Price Adjusment

H 0 = 0, H > 0

H

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p = H D p - S p

ExampleGeneral Price Adjusment

H 0 = 0, H > 0

iff D p* - S p* = 0 H D p* - S p* = 0

F p

F p = H D p - S p D p - S p

pWe studied the stability of F p

+ +-

< 0

Any price equilibrium is stable

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The Solow model of growth

λt

0

X t = F K t ,L t

K t = sX t

L t = L e

a production function, homogeneous of degree F 1

F K,L = LF K/L,1

define : k = K/L, f k = F k,1

k K L= -

k K L

sF K,L= - λ

K

sLF K/L,1= - λ

K sf k

= - λk

k

k

k = sf k - λk

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The Solow model of growth

k = sf k - λk

capital per worker

product per worker

rate of growth of number of workers

rate of saving

k =

f k =

λ =

s =

f 0 = 0, f k > 0, f k < 0

lim lim

Inada conditions :

k 0 k f k = , f k = 0

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The Solow model of growth

k = sf k - λk

define G k = sf k - λk

sf k - λk

k

G 0 = sf 0 - λ =

G = sf - λ = -λ

G k = sf k - λ

G k = sf k < 0

k*

= G k

is concaveG

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The Solow model of growth

k = sf k - λk

sf k - λk

kk*

= G k

k* is a unique stationary point, it is globally stable.

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Second Order Differential Equations

x = F t, x, x

2

2

d x d dxx = =

dt dt dt

The simplest possible equation of this type is:

x = k

by integrating : x = kt + A

integrating (2) : 2k x = t + At + B

2 for arbitrary A,B

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consider the equation : x = F t, x F t, x, x

change the variable : y = x

y = F y , t,This is a first order equation.

Example:x = x + t

y = y + t x y =

,-t -t tye ' = te y = x = Ae - t - 1

t 21x = Ae - t - t + B

2