The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x ' = f ( x ) with critical points at A ( − 1 , 0 ) , B ( 0 , 0 ) , C ( 0 , 1 ) are shown in the figure below:

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Answer 1

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Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

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Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.

************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.

Prove that Σ prime p≤x p=3 (mod 10) 1 Ρ = for some constant A. log log x + A+O 1 log x "

Answer 2

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

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Answer 3

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

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Answer 4

Question

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

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Answer 5

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Answer 6

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

Answer 7

Chapter 3.6, Problem 23P

(a)

To determine

The classification of each critical points according to the type (node, spiral point, saddle point, or center)and stability(unstable, stable or asymptotically stable ) from the direction field and the phase portrait for a certain two dynamical system x'=f(x) with critical points at A(−1,0) ,B(0,0) ,C(0,1) are shown in the figure below:

Answer 8

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Answer 9

(b)

To determine

limt→∞ ϕ(t) for each of the following set of initial conditions

i) x0=(0,0.1)T

ii) x0=(0,−0.1)T

iii) x0=(2,−1.5)T

iv) x0=(−2,1.5)T

Where the solution of the initial value problem x'=f(x), x(0)=x0 is denoted by ϕ(t)

with critical points at A(−1,0) ,B(0,0) ,C(0,1) and the direction field and the phase portrait are shown in the figure below:

Answer 10

Expert Solution & Answer

Answer 11

Expert Solution & Answer

Answer 12

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Answer 13

Solution Summary: The author explains that the critical point B(0,0) is an unstable saddle point.

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Chapter 3 Solutions