lecture 01 nus ma3252

Course Information Introduction Historical Problems

MA3220 Lecture 01

August 14, 2012

MA3220 Lecture 01


MA3220 Ordinary Differential Equations Instructor: Wong Yan Loi, S17-06-04, [email protected]

MA3220 Lecture 01


MA3220 Ordinary Differential Equations Instructor: Wong Yan Loi, S17-06-04, [email protected] Lecture: Tue, Fri, 2-3:30PM

MA3220 Lecture 01


MA3220 Ordinary Differential Equations Instructor: Wong Yan Loi, S17-06-04, [email protected] Lecture: Tue, Fri, 2-3:30PM Venue: LT33

MA3220 Lecture 01


MA3220 Ordinary Differential Equations Instructor: Wong Yan Loi, S17-06-04, [email protected] Lecture: Tue, Fri, 2-3:30PM Venue: LT33 Tutorial: 1 hour per week

MA3220 Lecture 01


MA3220 Ordinary Differential Equations Instructor: Wong Yan Loi, S17-06-04, [email protected] Lecture: Tue, Fri, 2-3:30PM Venue: LT33 Tutorial: 1 hour per week Brief lecture notes and tutorial questions are on IVLE

MA3220 Lecture 01


Prerequisites and Preclusion

Prerequisites: (MA1104 or MA1104S or MA1506) and(MA2108 or MA2108S) and (MA1101)Preclusion: MA2312, PC2174, FASS students from 2003 cohortonwards who major in Mathematics (for breadth requirement).

MA3220 Lecture 01


Tutorial

There are 8 tutorial sessions available.

Please register one tutorial group on CORS. Tutorial classeswill commence on the 3rd week of the semester.

MA3220 Lecture 01


Assessment

Assessment Final Exam: Friday, 30-Nov-2012, Afternoon session 70%

MA3220 Lecture 01


Assessment

Assessment Final Exam: Friday, 30-Nov-2012, Afternoon session 70% Mid term test: Tuesday, 9 Oct 2012, 2:00-3:30PM, Venue:

MPSH1-Section B, Tutorial 1-5 25%

MA3220 Lecture 01


Assessment

Assessment Final Exam: Friday, 30-Nov-2012, Afternoon session 70% Mid term test: Tuesday, 9 Oct 2012, 2:00-3:30PM, Venue:

MPSH1-Section B, Tutorial 1-5 25% Tutorial: 5%

MA3220 Lecture 01


Assessment

1. Students are allowed to bring along 2 sheets (4 sides) of A-4sized hand-written notes for the final exam and mid-term test.

2. The 5% of tutorial marks will be awarded if a student attainsEITHER at least 80% of attendance OR manages to work outcorrectly 2 problems in class + at least 60% of attendance.

3. Make-up test for those missing the mid-term test with validreason will be held on the 13rd week. (Tutorial 1 to 10).

MA3220 Lecture 01


References

Main reference: Differential Equations with Applicationsand Historical Notes by G.F.Simmons, 2nd edition,McGraw Hill.

MA3220 Lecture 01


References


An Introduction to Ordinary Differential Equations byE.A.Coddington, Dover 1961.

MA3220 Lecture 01


References



Essentials of Ordinary Differential Equations by R.P.Agarwal and C. Gupta, McGraw Hill 1991.

MA3220 Lecture 01


References



Essentials of Ordinary Differential Equations by R.P.Agarwal and C. Gupta, McGraw Hill 1991.

Differential Equations, by J. Polking, A. Boggess and D.Arnold, 2nd edition, Pearson Prentice Hall 2006.

MA3220 Lecture 01


Course Contents

First Order ODE

MA3220 Lecture 01


Course Contents

First Order ODE The theory of Linear ODE

MA3220 Lecture 01


Course Contents

First Order ODE The theory of Linear ODE System of ODEs

MA3220 Lecture 01


Course Contents

First Order ODE The theory of Linear ODE System of ODEs Series Solutions of 2nd order ODE

MA3220 Lecture 01


Course Contents

First Order ODE The theory of Linear ODE System of ODEs Series Solutions of 2nd order ODE Existence and Uniqueness Theorems

MA3220 Lecture 01


ODE

An ordinary differential equation (ODE) is a relation connectingthe function y of an independent variable x and its derivativesy , y , y , , y (n).

Examples:y = y2 + cos(xy)

y = 4x(y )2 + sin x

y + 2y + y = ex

MA3220 Lecture 01


Newton

Newton (1671): The method of fluxions and infinite series y = f (x) or y = f (y)

MA3220 Lecture 01


Newton

Newton (1671): The method of fluxions and infinite series y = f (x) or y = f (y) y = f (x , y)

MA3220 Lecture 01


Newton

Newton (1671): The method of fluxions and infinite series y = f (x) or y = f (y) y = f (x , y) First order partial differential equations

xzx + yzy = 2z

MA3220 Lecture 01


IsochroneJames Bernoulli (1690 in Acta Eruditorum) solved the problemof determining the isochrone:To find a curve in a vertical plane with the property that a heavypoint sliding without friction along this curve has constantvertical component of its velocity.

O x

y

b

1 +(dx

dy

)2= V2(2gy + v20 ),

where v0 is the initial speed, V is the constant vertical speed.MA3220 Lecture 01


Isochrone

By the Energy Conservation Law, the speed v(y) satisfiesv2 = 2gy + v20 . Let be the angle between the tangent to thecurve and the vertical direction. Then the vertical component ofthe speed is v cos = V . Eliminating v , we get

V 2 cos2 = 2gy + v20 .

Also tan = dxdy . Using cos2 = (1 + tan2 ), we get

1 +(dx

dy

)2= V2(2gy + v20 ).

MA3220 Lecture 01


Isochrone

Solution:

x(y) = V2

3g

(2gyV 2 +

v20V 2 1

) 32

+ C

is a semi-cubical parabola.

O x

y

b

MA3220 Lecture 01


Brachistochrone

Johann Bernoulli (1696) posed and solved the problem ofBrachistochrone (quickest descent)To find the curve connecting 2 points A and B that do not lie ona vertical line and possessing the property that a movingparticle slides down the curve from A to B in the shortest time.

A

B

x

y

b

b

b

MA3220 Lecture 01


Brachistochrone

Snells LawFor light travel through 2 mediums in least possible time, wemust have

sin1v1

=sin2

v2.

1

2

MA3220 Lecture 01


Brachistochrone

1

2

a

b

c

x c x

T =

a2+x2

v1+

b2+(cx)2

v2.

dTdx = 0 xv1

a2+x2

= cxv2

b2+(cx)2.

sin 1v1 =sin 2

v2.

MA3220 Lecture 01


Brachistochrone

For a continuous medium of decreasing density, we should alsohave

sinv

= constant.

v

b

MA3220 Lecture 01


BrachistochroneAs

sin = 11 + y 2

and v =

2gy ,

we have y(1 + (y )2) = c. Brachistochrone

Using the substitution tan(/2) =(

ycy

) 12, the solution is

x = c2( sin ), y = c2(1 cos ), a cycloid - the locus of apoint on the circumference of a circle as it rolls along a straightline. cpiO x

y

b

c2

MA3220 Lecture 01


The Motion of a Pendulum

ad2dt2 = g sin ,

For small , ad2

dt2 = g. a

m

Solution. Simple Harmonic Motion:

= cos

(ga

t

).

MA3220 Lecture 01


Using d2

dt2 = d d , and integrating the original equation, we

havedtd =

a

2g1

cos cos.

Hence, Period T = 2

ag

0

dk2 sin2(/2)

, k = sin(/2),

which is an elliptic integral!

MA3220 Lecture 01


(1695) The Bernoulli Equation: dydx + P(x)y = Q(x)yn,

n = 0,1,2, (Solved by the change of variable z = y1n.)(1712) The Riccati Equation: An extension of y = p(x) + q(x)yis y = p(x) + q(x)y + r(x)y2.

(1829) The Abel Equation: An extension of Riccatis Equation isy = p(x) + q(x)y + r(x)y2 + s(x)y3.

(1728) L. Euler solved the Riccati equation by the substitutiony = y1 + 1/z, where y1 is a particular solution of the Riccatiequation.

Euler invented the method of variation of parameters which waselevated by a general procedure by Lagrange in 1774.

MA3220 Lecture 01


Legendres Equation (1 x2)y 2xy + p(p + 1)y = 0.

Hermites Equation y 2xy + 2py = 0.

Laguerres Equation xy + (1 x)y + y = 0.

Chebyshevs Equation (1 x2)y xy + p2y = 0.

Bessels Equation x2y + xy + (x2 p2)y = 0.

Gausss Hypergeometric Equationx(1 x)y + (c (a + b + 1)x)y aby = 0.

Airys equation y + xy = 0.

MA3220 Lecture 01


Applications

Applications: Chemical kinetics, population growth, economics,mathematical finance, radioactive decay, isoperimetricproblems, dynamical systems, chao etc.

Mathematicians: DAlembert, Clairaut, Cauchy, Bernoulli,Picard, Lipschitz, Laplace, Legendre, Euler, Poisson, Peano,Poincare etc.

MA3220 Lecture 01

Course InformationIntroductionHistorical Problems

lecture 01 nus ma3252

Documents

weekma3220 lecture

ivlema3220 lecture

lt33ma3220 lecture

2012ma3220 lecture

30pmma3220 lecture

week brief lecture notes

lt33 tutorial

tutorial questions