Maths Assignment 1 - 2C Feedback Exercise 2018-19 Solutions

This document provides a detailed solution to a Maths assignment focusing on a population model. The assignment involves analyzing the equilibrium points of a given differential equation, sketching the function and phase portrait, and classifying the stability of the equilibrium points. The solution includes calculations, graphical representations, and discussions on the behavior of the population model over time with different initial conditions. The assignment also includes questions involving the analysis of parametric equations, and sketching the curves represented by these equations. The solution demonstrates the steps to find the relationship between the variables, sketch the curves, and find the range of the parameters. Furthermore, the assignment requires the use of mathematical concepts such as parametric equations, and trigonometric functions.

Maths assignment 1
MATHS ASSIGNMENT
By Name
Course
Instructor
Institution
Location
Date

Maths assignment 2
QUESTION ONE
Given
σN
σt = f(N)
∝(N –m1)(N–m2) ( 1– N
m3 )
∝(N –m) ( 1– N
m3 ¿2
(1- N
m3 ¿=0
From the above graph it is clear that NR=m3 is an unstable equilibrium solution N=m is an
asymptotically stable equilibrium solution.
 N=m2 be lower differently from these two
 The equilibrium solution semi stable
 The starts above it toward
 N=m2 the start below

Maths assignment 3
 N=m2 move away as (t) increase.
QUESTION TWO
Part a
Here a and b are not given explicitly
We suppose a= i and b = j
X = ( 1-t/2) i+j
Let X = ( ni + yj)
Where n = 1-t/2 , y =t
And n= 1- y
2
2n+y=2
0≤ t ≤ 2
Therefore 0≤ n ≤1 and 0≤ y ≤ 2

Maths assignment 4
Part b
Here, n=u2 and y=u
n=y2
y2=n
-2≤ u≤ 2
-2≤ y ≤ 2 and 0 ≤ n ≤ 4
Part C

Maths assignment 5
n= t
2 π Cost
y= t
2 π Sint
n2+y2= t2
2 π2 and
tant = y
n
t= tan-1 y
n
n2+y2 = 1
4 π2 (tan-1 y
n ¿2
tan-1 y
n = 2π √n2+ y2
If n=rcosθ , y=r sinθ , then ,
2π √r 2 cos2 θ+r 2 sin2 θ = tan-1 tanθ
2 πr=θ
r= θ
2 π
now, t= tan-1 y
n =0

Maths assignment 6
Bibliography
Kamal, A. A., 2011. 1000 Solved Problems in Classical Physics: An Exercise Book. 3rd ed.
Stoke: Springer Science & Business Media.
Morin, D., 2008. Introduction to Classical Mechanics: With Problems and Solutions. 2nd ed.
London: Cambridge University Press.

Maths assignment 7