This course is replaced by MAT3440 – Dynamical systems.

Undervisningsplan

DatoUndervises avStedTemaKommentarer / ressurser
16.01.2012Terje Sund (TS)  Kristen Nygaards hus 2116 (Store aud)  Edwards & Penney (EP) 1.1, 1.2, 1.4  First-order differential equations and mathematical models. Separable equations. 
18.01.2012TS  Kristen Nygaards 2116 Store aud (KN 2116)  EP 1.3, 1.5  Slope fields. Existence and uniqueness. Linear equations  
23.01.2012TS  KN 2116  EP 1.6, ( 2.1)  Substitution methods and exact equations. (Second-order linear equations.) 
25.01.2012TS  KN 2116  Exercises (Problem Session) and Section 1.6: Reducible equations.  Exercises 1.1.10, 1.1.30, 1.1.45, 1.2.18, 1.4.5, 1.4.18, 1.4.35, 1.4.53, 1.4.66

EP Section 1.6: Reducible equations

 

30.01.2012TS  KN 2116  EP 2.1, (2.2 Theorem 5, 2.3 Theorem 3)  Second-order linear equations. General solutions. Wronskians. Uniqueness. Characteristic equations. Nonhomogeneous linear equations (emphasis on equations of second order). 
01.02.2012TS  KN 2116  Exercises  In EP

1.3: 1, 11, 12, 16, 23, 30, (32)

1.5: 22, 25, (31)

1.6: 1, 5, (57, 58, 62)

Exercises in parenthesis will only be done if there is enough time. 

06.02.2012TS  KN 2116  EP 2.2 Theorem 5, 2.3 Theorem 3, 2.5  Repeated roots, complex roots. Nonhomogeneous linear equations. Undetermined coefficients, with emphasis on equations of second order. Variation of parameters. 
08.02.2012TS  KN 2116  Exercises  In EP

2.1: 1, 17 21, 24, 32, 33, 38, 40, 43, 51, 52.

1.6: 57, 58, 62 

13.02.2012TS  KN 2116   SSS 4.5 (pp.114-116, first half of p.117).11.1, 11.2 (SSS=Sydsæter, Seierstad, Strøm, Matematisk Analyse II)  Convex and concave functions and the 2nd-derivative test. Calculus of Variations. The Euler equation. The main theorem. 
15.02.2012TS  KN 2116   Exercises  In EP

2.2: 25, (26), 36, 44

2.3: 22, 23

2.5: 1, 3, 4, 57 

20.02.2012TS  KN 2116  SSS 11.2-11.5  Calculus of Variations. Proof of the Main Theorem 
22.02.2012TS  KN 2116  Exercises  In EP:

2.5: 10, 61

In SSS:

4.5: 2, 3

11.2: 1, 7, 8, 9

(11.3: 1, 2, 3 except the case T=1.) SOLUTIONS

 

27.02.2012TS  KN 2116  SSS 11.3-11.5. EP 5.1  The case of a free right endpoint (in calculus of variations).

First-order linear systems of differential equations. 

29.02.2012TS  KN 2116  Exercises  In SSS (text in Norwegian)

11.3: 1, 2, 3 except the case T=1. 11.5: 2 (a) and (b).

Exam June 2006, Problem 1 (Oppgave 1) 

05.03.2012TS  KN 2116  EP 5.1-5.3, 5.4  The method of elimination. Matrices and linear systems. The eigenvalue method. 
07.03.2012TS  KN 2116  Exercises. Theory from 5.3 and 5.4  EP

5.1: 8, 21; 5.3: 21; 5.4: 17, (24)

+ General solutions of linear first order systems. The eigenvalue method. 

12.03.2012TS  KN 2116  EP 5.3, 5.4, (5.6)  Wronskians of systems. Solutions of nonhomogeneous systems. Complex eigenvalues. (Multiple eigenvalue solutions.)

Obligatory Problem Set 

14.03.2012TS  KN 2116  Exercises. Theory from 5.6.  EP

5.4: 24, 20; 5.6: 1, (2) (you may use the method of elimination on 5.6: 1 and 2)

From 5.6: Multiple eigenvalue solutions.(Defective eigenvalues.) 

19.03.2012TS  KN 2116  EP 5.6  Multiple eigenvalue solutions. Defective eigenvalues.  
21.03.2012TS  KN 2116  Exercises  5.6: 2, 3, 7, 15, 16, (21) 
26.03.2012TS  KN 2116  EP 5.7   Matrix exponentials.  
28.03.2012TS  KN 2116  Cancelled (because of midterm exams)   

April 2- April 10: Easter vacation.

DatoUndervises avStedTemaKommentarer / ressurser
11.04.2012TS  KN 2116  EP 5.7, 5.8  Matrix exponentials. Variation of parameters.  
16.04.2012TS  KN 2116   SSS 12.1-12.4  Optimal Control Theory. The Maximum Principle (first version). Mangasarian's Theorem (first version). Examples.  
18.04.2012TS  KN 2116  Exercises  EP 5.7: 3, 21, 25, 29, 34, 37. (5.8: 19) 
23.04.2012TS  KN 2116  SSS 12.3-12.4   Optimal Control Theory. The Maximum Principle (general version). Mangasarian's Theorem (general version). Arrow's condition. 
25.04.2012TS  KN 2116  Exercises  SSS 12.2: 1, 2, 3. EP 5.8: 2, 24 
30.04.2012TS  KN 2116  EP 7.1-7.3  Nonlinear systems. Almost linear systems, linearization. Equilibrium solutions and stability. The phase plane: Real eigenvalues. 
02.05.2012TS  KN 2116  Exercises   SSS 12.4: 1, 2, 3 (EP 7.2: 9. Also find the linearizations at the critical (equilibrium) points. Determine the type of each critical points.)  
07.05.2012TS  KN 2116  EP 7.2, 7.3   Linear and almost linear systems. Linearization. Equilibrium solutions and stability. The phase plane: Complex eigenvalues. 
09.05.2012TS  KN 2116  Exercises  SSS 12.4.6 (b). 12.7.1. Exam June 2006, Oppgave 2.

(Exam June 2010, Oppgave 1 )

 

14.05.2012TS  KN 2116  EP 6.1, 6.2, 6.3  Euler's Method. The improved Euler Method. The Runge-Kutta Method. 
16.05.2012TS  KN 2116  EP 6.3. Exercises  EP 7.3: (17,) 29, 30, 33. Exam June 2010, Oppgave 1 Notice that likevektspunkt = critical point (also called equilibrium point.)

The following: Problem

Solution to Problem 

21.05.2012TS  KN 2116  Exercises  Exam 2011 
Published Jan. 2, 2012 8:16 PM - Last modified May 23, 2012 6:22 PM