Transcript
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 1/54
Honours Finance (Advanced Topics in Finance:Nonlinear Analysis)
Lecture 2: Introduction toOrdinary
Differential EquationsBY:
Abdul Ghani KhanM.tech
School of ICT,GBU
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 2/54
W hy bother?Last week we considered Minsky·s Financial Instability
Hypothesis as an expression of the ´endogenousinstabilityµ explanation of volatility in finance (andeconomics)The FIH claims that expectations will rise during periods
of economic stability (or stable profits).That can be expressed as² % rate of change of expectations = f(rate of growth),
or in symbols
¹ º ¸©
ª¨ v!v
dt dY
Y f
dt dE
E 11
This is an ordinary differential equation (ODE); exploringthis model mathematically (in order to model it) thus
requires knowledge of ODEs
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 3/54
W hy bother?In general, ODEs (and PDEs) are used to model real-life
dynamic processes² the decay of radioactive particles² the growth of biological populations² the spread of diseases² the propagation of an electric signal through a circuitEquilibrium methods (simultaneous algebraic equations
using matrices etc.) only tell us the resting point of areal-life process if the process converges to equilibrium
(i.e., if the dynamic process is stable)Is the economy static?
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 4/54
Economies and economic methodologyEconomy clearly dynamic, economic methodology primarily
static. W hy the difference?Historically: the KISS principle:² ´If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficultquestion when the more easy one is yet so imperfectlywithin our power.µ (Jevons 1871 [1911]: 93)
² ´...dynamics includes statics... But the staticalsolution« is simpler...; it may afford useful preparationand training for the more difficult dynamical solution;and it may be the first step towards a provisional andpartial solution in problems so complex that a completedynamical solution is beyond our attainment.µ
(Marshall, 1907 in Groenewegen 1996: 432)
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 5/54
Economies and economic methodologyA century on, Jevons/Marshall attitude still dominates most
schools of economic thought, from textbook to journal:² Taslim & Chowdhury, Macroeconomic Analysis for Australian
Students: ´the examination of the process of moving fromone equilibrium to another is important and is known asdynamic analysis. Throughout this book we will assume thatthe economic system is stable and most of the analysis willbe conducted in the comparative static mode.µ (1995: 28)
² Steedman, Questions for Kaleckians: ´The general pointwhich is illustrated by the above examples is, of course, that
our previous 'static' analysis does not 'ignore' time. To thecontrary, that analysis allows enough time for changes inprime costs, markups, etc., to have their full effects.µ(Steedman 1992: 146)
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 6/54
Economies and economic methodologyIs this valid?² Yes, if equilibrium exists and is stable² No, if equilibrium does not exist, is not stable, or is one of
many...Economists assume the former. For example, Hicks on Harrod:
² ´In a sense he welcomes the instability of his system,because he believes it to be an explanation of the tendencyto fluctuation which exists in the real world. I think, as Ishall proceed to show, that something of this sort may wellhave much to do with the tendency to fluctuation. But
mathematical instability does not in itself elucidatefluctuation. A mathematically unstable system does notfluctuate; it just breaks down. The unstable position is one inwhich it will not tend to remain.µ (Hicks 1949)
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 7/54
Lorenz·s Butterfly
So, do unstable situations ´just break downµ?² An example: Lorenz·s stylised model of 2D fluid flow
under a temperature gradientLorenz·s model derived by 2nd order Taylor expansion
of Navier-Stokes general equations of fluid flow. Theresult:
dx a y x
dt dy
b z x y dt dz
x y c z dt
! v
! v
! v v
x displacement
y displacement
temperature gradient
Looks pretty simple, just a semi-quadratic«First step, work out equilibrium:
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 8/54
Lorenz·s Butterfly
Three equilibria result (for b>1):
00
0
d x y x
d t d y
z x y d t d z
x y c z d t
! v !
! v !
! v v !
,1, 1
1
y x a
z b b
x y b c
x y z
! {! "
! ! v! ! !
111
b c x
y b c
z b
« » v« » ¬ ¼¬ ¼ ¬ ¼! v¬ ¼ ¬ ¼¬ ¼ ¬ ¼- ½ ¬ ¼- ½
000
x
y
z
« » « »¬ ¼ ¬ ¼!¬ ¼ ¬ ¼¬ ¼ ¬ ¼- ½ - ½
11
1
b c x
y b c
z b
« » v« » ¬ ¼¬ ¼ ¬ ¼! v¬ ¼ ¬ ¼¬ ¼ ¬ ¼- ½
¬ ¼- ½
Not so simple after all! Buthopefully, one is stable andthe other two unstable«Eigenvalue analysis gives the
formal answer (sort of «)But let·s try a simulation
first «
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 9/54
Simulating a dynamic system
Many modern tools exist to simulate a dynamic system² All use variants (of varying accuracy) of approximation
methods used to find roots in calculusMost sophisticated is 5th order Runge-Kutta; simplest
Euler
² The most sophisticated packages let you see simulationdynamically
W e·ll try simulations with realistic parameter values,starting a small distance from each equilibrium:
5151
a
b
c
« » « »¬ ¼ ¬ ¼!¬ ¼ ¬ ¼¬ ¼ ¬ ¼- ½ - ½
So that theequilibria are
3.742 3.742
03.742 , 3.742 , 0 14 14 0
x y
z
« » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼! ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼- ½ - ½ - ½ - ½
Lorenz_Any.vsm
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 10/54
Lorenz·s ButterflyNow you know where the ́ butterfly effectµ came from² Aesthetic shape and, more crucially
All 3 equ i l ibria ar e unst ab le (shown later)² Probability ze ro that a system will be in an equilibrium
state (Calculus ´Lebesgue measureµ)Before analysing why, review economists· definitions of
dynamics in light of Lorenz:² Textbook: ´the process of moving from one equilibrium
to anotherµ. W rong:
² system starts in a non-equilibrium state, and movesto a non-equilibrium state² not equilibrium dynamics but far-from equilibrium
dynamics
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 11/54
Lorenz·s Butterfly² Founding father: ´mathematical instability does not in
itself elucidate fluctuation. A mathematically unstablesystem does not fluctuate; it just breaks downµ.W rong:
System with unstable equilibria does not ´break downµbut demonstrates complex behaviour even withapparently simple structureNot breakdown but complexity
² Researcher: ́ static « analysis allows enough time forchanges in prime costs, markups, etc., to have theirfull effectsµ. W rong:
Complex system will remain far from equilibrium even ifrun for infinite timeConditions of equilibriumneve r relevant to systemic
behaviour
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 12/54
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 13/54
Nonlinearities in economicsStructural² monetary value of output the product of price and
quantityboth are variables and product is quasi-quadratic
Behavioural² ´Phillips c u r ve µ relation
wrongly maligned in literatureclearly a curve, yet conventionally treated as linear
Dimensions² massively open-multidimensional, therefore numerous
potential nonlinear interactionsEvolution² Clearly evolving system, therefore even more complex
than ´simpleµ nonlinear dynamics«
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 14/54
W hy bother?
0 10 20 30 40 50 60 70 80 90 100
Time
-8
-6
-4
-2
-0
2
4
6
8
X w
i t h
b =1
5
Lorenz's "strange attractor "X displacement as a function of time
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 15/54
W hy Bother?Lorenz·s bizarre graphs indicate² Highly volatile nonlinear system could still be
systemically stablecycles continue forever but system never exceeds
sensible bounds
² e.g., in economics, never get negative priceslinear models however do exceed sensible bounds² linear cobweb model eventually generates negative
prices² Extremely complex patterns could be generated by
relatively simple modelsThe ´kissµ principle again: perhaps complex systems
could be explained by relatively simple nonlinearinteractions
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 16/54
W hy Bother?But some problems (and opportunities)² systems extremely sensitive to initial conditions and
parameter values² entirely new notion of ´equilibriumµ
´Strange attractorsµ
² system attracted to region in space, not a pointMultiple equilibria² two or more strange attractors generate very
complex dynamics
² Explanation for volatility of weatherEl Nino, etc.
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 17/54
W hy bother?
0 10 20 30 40 50 60 70 80 90 100
Ti e
-8
-6
-4
-2
-0
2
4
6
8
X d
i
p l a e
e n t
Loren ' Strange Attra tor One small step f or a butter f ly, one enor mous f lap f or mankin d... Tiny error
in initialreadings
leads toenormousdifferencein time pathof system.And behindthe chaos,
strangeattractors...
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 18/54
W hy bother?
Loren ' s trange ttra tor ,Y an d Z displa e ment
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 19/54
W hy Bother?Lorenz showed that real world processes could have
unstable equilibria but not break down in the long runbecause² system necessarily diverges from equilibrium but does
not continue divergence far from equilibrium
² cycles are complex but remain within realistic boundsbecause of impact of nonlinearitiesDynamics (ODEs/PDEs) therefore valid for processes
with endogenous factors as well as those subject to anexternal force² electric circuit, bridge under wind and shear stress,
population infected with a virus as before; and also² global weather, economics, population dynamics with
interacting species, etc.
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 20/54
W hy Bother?To understand systems like Lorenz·s, first have to
understand the basicsDifferential equations² Linear, first order² Linear, second (and higher) order² Some nonlinear first order² Interacting systems of equationsInitial examples non-economic (typical maths ones)Later we·ll consider some economic/finance applications
before building full finance model
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 21/54
Maths and the real worldMuch of mathematics education makes it seem irrelevant
to the real worldIn fact the purpose of much mathematics is to
understand the real world at a deep levelPrior to Poincare, mathematicians (such as Laplace)
believed that mathematics could one day completelydescribe the universe·s futureAfter Poincare (and Lorenz) it became apparent that to
describe the future accurately required infinitelyaccurate knowledge of the present² Godel had also proved that some things cannot be
proven mathematically
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 22/54
Maths and the real worldToday mathematics is much less ambitiousLimitations of mathematics accepted by most
mathematiciansMathematical models² seen as ´first passµ to real world² regarded as less general than simulation models
but maths helps calibrate and characterise behaviour ofsuch models
² ODEs and PDEs have their own limitations
most ODEs/PDEs cannot be solved² however techniques used for those that can are used
to analyse behaviour of those that cannot
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 23/54
Maths and the real worldSummarising solvability of mathematical models (from
Costanza 1993: 33):
Linear Non-linear
EquationsOneequation
Severalequations
Manyequations
Oneequation
Severalequations
Manyequations
Algebraic trivial easyessentiallyimpossible very difficult very difficult impossible
OrdinaryDifferential easy difficult
essentiallyimpossible very difficult impossible impossible
PartialDifferential difficult
essentiallyimpossible impossible impossible impossible impossible
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 24/54
Maths and the real worldTo model the vast majority of real world systems that
fall into the bottom right-hand corner of that table, we² numerically simulate systems of ODEs/PDEs² develop computer simulations of the relevant processBut an understanding of the basic maths of the solvable
class of equations is still necessary to know what·s goingon in the insoluble set² Hence, a crash course in ODEs, with some refreshers
on elementary calculus and algebra...
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 25/54
From Differentiation to Differential«In Maths 1.3, you learnt to handle equations of the form
x f d xdy !
W here f is some function. Forexample
c x y
d x xd xd xd y
x
d x
d y
!
!
!
´´cos
sin
sin
On the other hand, differential equations are of the
form y x f
dxdy
,!So how do we handle them? Make them look like the
stuff we know:
The rate of change of y is a function of itsvalue: y both independent & dependent
Dependent variable Independent variable
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 26/54
From Differentiation to Differential«The simplest differential equation is
ydt d y ! (we tend to use t to signify time , rather than x
for displa cement as in simple differentiation)
Trysolvingthis for
yourself:
ct dt ydt y
y
dt d y y y
ydt d y
ydt d y
!!!
!
v!
!
!
´´1lnln
dt
d
:tw.r.t.sides bothIntegrate1lndtd
:ormin thisequationtheRewrite1lndtd
:A tri ck 1
y bysides bothDivide
Continued...
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 27/54
From Differentiation to Differential«
Another approach isn·t quite so
formal:
growthlExponentia
lsexponentiaTa k eln
t
t
t ct ct
eC y
C e y
C eeee y
ct y
v!
!v
v!v!!!B ecause log of a negative
number is not definedB ecause an
exponential isalways positive
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 28/54
From Differentiation to Differential«Treat dt as a small quantityMove it around like a variableIntegrate both sides w.r.t the relevant
´d(x)µ term² dy on LHS² dt on RHSSome problems with generality of this
approach versus previous method, but OKfor economists & modelling issues
t e yct y
dt yd y
dt y
d y
ydt d y
v!!
!!
!
´´ln
So what·s the relevance of this toeconomics and finance? How aboutcompound interest?
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 29/54
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 30/54
From Differential Equations to FinanceUnder what circumstances will our moneylender·s assets
grow?² C equals his/her initial assets:
t p pi set y vvv!
C C eC eC y p pi s !v!v!v! vv 10 00
The moneylender will accumulate if the power of theexponential is greater than zero:
gpgp"!v v¹ º ¸©ª̈ tasethen0If t p pi s PP
The moneylender will blow the lot if the power of theexponential is less than zero:
gppv
tas0ethen0It P
P
Known as ´eigenvalueµ;Known as ´eigenvalueµ;tells how much the equationtells how much the equation
is ´stretchingµ spaceis ´stretchingµ space
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 31/54
Back to Differential Equations!The form of the preceding equation is the simplest
possible; how about a more general form:
yt f d t
dy v!Same basic idea applies:
dt t f yd y v!
´ v! dt t f yln
´ vv! dt t f eC y
f(t) can take many forms, and all your integrationknowledge from Maths 1.3 can be used« A few
examples
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 32/54
Back to Differential Equations!But firstly a few words from our sponsor² These examples are just ´roteµ exercises
most of them don·t represent any real world system² However the ultimate objective is to be able to
comprehend complex nonlinear models of finance that
do purport to model the real worldso put up with the rote and we·ll get to the finalobjective eventually!
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 33/54
Back to Calculus!
Try the following:
0sin
0
0sin
0
0
2
!vv
!v
!v
!v
!v
v
v
yt edt d y
ye
dt
d y
yt dt d y
yt dt d y
yt dt d y
t b
t b
W on¶t pursue the last one be causeN ot a course in integrationM ost differential equations
analyti cally insoluble any wayP rograms exist which can do most
( but not all!) integrations a human cando
B ut a qui ck reminder of what is doneto solve su ch O Es
A lso of relevan ce to wor k we¶lldo later on systems of O Es
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 34/54
Back to Calculus!
Simple to derive from first principles: consider afunction which is the product of two other functions:
ud xd
vvd xd
uvud xd vv!v
Some useful rules from differentiation and integration:² Product rule:
t et f t b sinv! v
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 35/54
Back to Calculus!
10
10
u x( )
x( )
u x( ) x( )
1010 x10 5 0 5 10
10
5
0
5
10
s¡
¢ (x)exp (-bx)s
¡
¢ (x)*e xp (-bx)
dxd u
vdxd v
u
x
uv x
vu
xvu
xvuuvvuvuvu
x
vuvvuu
x f f f
x f dxd
vvuu f f vu f
x
x
x
x
vv!
((v(
(v(
((v!
(v(v(v((vv!
(
v(v(!
((!
(v(!(v!
p(
p(
p(
p(
lim
lim
lim
lim
0
0
0
0
Thenxof functionsarevu,f ,whereConsider
These rules thenreworked to give us´integration by partsµ forcomplex integrals:
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 36/54
Back to Calculus!
Convert difficult integration into an easier one by either² reducing ´uµ component to zero by repeateddifferentiation
² repeating ´uµ and solving algebraically
´´ vv!vvv!vvv!v
vv!v
d uvvudvu
d uvvud dvu
d uvdvuvud d xd u
vd xdv
uvud xd
Treat integration as amultiplication operator
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 37/54
Back to Calculus!Practically
² choose for ´uµ something which eithergets simpler when integrated; orcycles back to itself when integrated more than once
² For our example:
t et f t b sinv! v
These don·t getany simpler, but
do ´cycleµ
T ry sin:cycles backformulas exist for
expansion
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 38/54
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 39/54
Back to Calculus!
Stage Two:
t f unc t ion
t f unc t iond t t e X
t f unc t ion X t f unc t ion
X t f unc t iont f unc t ion X
b
at b
ab
ba
!v!!v
v!
´ v
1)(
sin
)(1
)(
Finally, Stage Three: we were trying to solve the ODE:
0sin !vvv yt ed t dy t b
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 40/54
Back to Differential Equations!W e got to the point where the equation was in soluble
form:
´´ vv! v d t t e ydy t b sin
Then we solved the integral:
t f unc t ion
t f unc t iondt t e X
b
at b !v! ´ v
1)(
sin
Now we solve the LHSand take exponentials:
t f unc t ion
t f unc t ion
b
a
b
a
eC y
t f unc t iont f unc t ion y
v!
!
1
1ln
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 41/54
Back to Differential Equations!So far, we can solve (some) ordinary differential
equations of the form:
0!v yt f dt d y
These are known as:² First order
because only a first differential is involved² Linear
Because there are no functions of y such as sin(y)² Homogeneous
Because the RHS of the equation is zero
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 42/54
Back to Differential Equations!Next stage is to consider non-homogeneous equations:
t g yt f dt d y !v
g(t) can be thought of as a force acting on a systemW e can no longer ´divide through by yµ as before,since this yields
dt t f y
t g
y
d y v¹¹ º
¸©©ª
¨!
which still has y on both sides of the equals sign, andif anything looks harder than the initial equationSo we apply the three fundamental rules of
mathematics:
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 43/54
The three fundamental rules of mathematics
(1)W hat have you got that you don·t want?
² Get rid of it(2)W hat haven·t you got that you do want?² Put it in(3) Keep things balancedTake a look at the equation again
t g yt f dt d y !vW hat does this look almost like?
The product rule: ud xd
vvd xd
uvud xd vv!v
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 44/54
Non-Homogeneous First Order Linear ODEsThe LHS of the expression
t g yt f dt d y !v is almost in product rule
form
Can we do anything to putit exactly in thatform?² Multiply bo th s i d es by an expression Q(t)
so that t g t yt f t dt d y
t v!vvv Q Q Q
This is only possible if
yt f t dt d y
t yt dt d vvv!v Q Q Q
Now we have to find aQ(t) such that
t f t t d t d v! Q Q
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 45/54
The Integrating Factor ApproachThis is a first order linear homogeneous ODE, which we
already know how to solve (the only thing that makes itapparently messy is the explicit statement of adependence on t in Q(t), which we can drop for a while):
´ v!
v!
v!
v!v!
´́´d t t f et
d t t f
d t t f d
d t t f d
t f d t
d
Q
Q Q
Q Q Q
Q Q
lnThis is k nown as the ³integrating fa ctor´
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 46/54
The Integrating Factor ApproachSo if we
multiply t g yt f dt d y !v by ´ v! d t t f et Q we get
´v!¹ º ¸©ª
¨ ´v!v´v´ dt t f dt t f dt t f dt t f et g e y
dt d
yet f dt d y
e
Anybody dizzy yet?² It·s complicated, but there is a light at the
end of the tunnel
Next, we solve the equation by takingintegrals of both sides:
dt et g e ydt e ydt d dt t f dt t f dt t f ´´ ´v!´v!¹ º
¸©ª¨ ´v
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 47/54
The Integrating Factor ApproachAnd finally the solution is:
´
´v! ´
dt t f
dt t f
e
dt et g y
This is a bit like line dancing: it looks worse than itreally is.² Let·s try a couple of examples: firstly, try t yt
dt d y !vv2
(Actually, line dancing probably is as bad as it looks,and so is this)...
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 48/54
The Integrating Factor Approach
The first one t yt dt d y !vv2 becomes
t yt dt d y v!vvvv Q Q Q 2 using the integrating factor
Now we need aQ suchthat
yt dt d y y
dt d vvvv!v Q Q Q 2
Q Q vv! t dt d
2W hich is only possibleifThis is a first order homogeneous DE: piece of cake!
222ln2 t et dt t dt t d !p!vv!pvv! ´ Q Q Q Q
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 49/54
The Integrating Factor ApproachThus we multiply
t yt dt d y !vv2 by
2t e to yield
2222
2 t t t t et yedt d
et ydt d y
e v!v!vvvv
Then we integrate:
´´ v!v!v dt et yedt yedt d t t t 222
Next problem: how to integrate this?
Back to basics #2:the Chain Rule in
reverse
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 50/54
The Chain RuleThis expression:
´ v d t et t 2
´Looks likeµ t t dt d
t ud ue u v!!´ 2sincewhere 22
22 so2
1 t udt t t d !v!vOr in differential form:
That integral is elementary: ced ue uu !´
C eced ued uet t t ut v!v!v!v ´´ 222
21
21
21
Now substituting for u and taking account of theconstant:
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 51/54
The Integrating Factor ApproachFinally, we return to
´´ v!v!v dt et yedt yedt d t t t 222
Putting it all together: C e ye t t v!v 22
2
1
2
22
2
21
212
1t
t t
t
eC e
C
e
C e y v!!
v!
t yt dt d y !vv2
is the solution to
Before we try another example, the generalprinciple behind the technique above is the chainrule in reverse:
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 52/54
The Chain Rule
10
10
u x( )
£ x( )
u x( ) £ x( )( )
¤ u x( )( )
1010 x10 5 0 5 10
10
5
0
5
10
s i¥ (x)¦ xp (-bx)s i¥ (x)* ¦ xp (-bx)¦ (s i¥ (x))
C t g F
dt t g t g F dt t g t g f
x f x F d x x f x F
!v!v
!!
´´´
'''o
'thenI
In reverse, thesubstitution method ofintegration:
Rate of change ofcomposite function is rateof change of one times theother
x g x g f x F
x g f x F '''then
Iv!
!
=slope of composite
Slope of one
* slope of other
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 53/54
Back to Differential Equations!Try the technique with
12 !v yt d t dy
3
3
1
32
2
2
2
2
3
1dtln
d
dtddtd
iff
:factor gintegratinTimes1
t e
t t
t
t
yd t d yt
d t dy
yt d t dy
!
!v!!
v!
v!
!v!vvv
!v
´´ Q
Q Q Q
Q Q
Q Q
Q Q Q Q
Stage One: Finding Q:
8/7/2019 08 ODE Intro Ghani
http://slidepdf.com/reader/full/08-ode-intro-ghani 54/54
Linear First Order Non-HomogeneousStage Two: applyQ:
cdt e yedt yedt d
e yedt d
yet dt d y
e yt dt d y
e yt dt d y
t t t
t t t t
t
!v!¹¹ º
¸©©ª
¨v
!¹¹ º
¸©©ª
¨v!vvv!vvv
!!v
´´333
3333
3
3
13
13
1
3
13
13
123
12
3
12 actor gintegratinTimes1
Q Q
Q
Stage Three: integrating RHS«² there is no known integral! (common situation in
ODEs)² Completing the maths as best we can:
cedt ee yt t t
vv! ´333
3
13
13
1
This can only be estimated numerically
top related