APM 2614 ASSIGNMENT 01 questions with correct answers Chapters 1, 2 and 3

APM2614/201/1/2017




Tutorial letter 201/1/2017


APPLIED DYNAMICAL SYSTEMS
APM2614

Semester 1


Department of Mathematical Sciences


This tutorial letter contains solutions for assignment 01.




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university
Define tomorrow. of south africa

, SEMESTER 01
ASSIGNMENT 01
DUE DATE: 24 March 2017
UNIQUE ASSIGNMENT NUMBER: 879290
COMPULSORY ASSIGNMENT FOR THE EXAM, STUDY GUIDE: Chapters 1,
2 and 3


Question 1

(a) Singular point
2x − 4y = 0; x = 2y
3x − 5y + 2 = 0
6y − 5y + 2 = 0; y = −2
∴ sungular point (−4, −2)


2−4 σ = −3, ∆ = 2
A = ; 2
3−5 0 < ∆ < σ4
Singular point is a stable node


2−λ −4
= 0 ⇒ λ = −1 or − 2
3 −5 − λ
    
3−4 x 4
λ = −1 : =0 3x − 4y = 0 so the e − vector
3−4 y 3
    
4−4 x 1
λ = −2 : =0⇒x=y so the e − vector
3−3 y 1
       
x 4 1 −4
General solution = C1 e−t + C2 e−2t +
y 3 1 −2


(b) x+y =0
−x − y = 0 singular point not unique (x = −y)
 
1 1 1−λ 1
A= , =0
−1 −1 −1 −1 − λ

λ=0

  
1 1 x
e − vector =0 x + y = 0.
y



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