Explanation Given: The system of differential equations is, x ′ = − 2 x − y , y ′ = 3 x − y Approach: 1) Critical points are the solutions of x ′ = 0 and y ′ = 0 . 2) Trajectories are the parametric solutions of the system of differential equations. 3) Eigenvalues are the roots of characteristic equation. 4) Classification of critical points: i) For complex conjugates eigenvalues: • When the real part λ is zero, this type of critical point is called a center. It is neutrally stable. • When the real part λ is nonzero, this type of critical point is called a spiral point. It is asymptotically stable if real part is less than 0 , it is unstable if real part is greater than 0 . ii) For real eigenvalues: Distinct eigenvalues • When both eigenvalues are positive or both are negative, this type of critical point is called a node. It is asymptotically stable if eigenvalues are both negative, unstable if eigenvalues are both positive. • When eigenvalues have opposite signs, this type of critical point is called a saddle point. It is always unstable. Repeated eigenvalues • When there are two linearly independent eigenvectors, this type of critical point is called a proper node. It is asymptotically stable if eigenvalue is less than 0 and unstable if eigenavalue is greater than 0 . • When there is only one linearly independent eigenvector, this type of critical point is called an improper node. It is asymptotically stable if eigenvalue is less than 0 , unstable if eigenavalue is greater than 0 . Calculation: Solve x ′ = 0 − 2 x − y = 0 y = − 2 x Solve y ′ = 0 3 x − y = 0 3 x − ( − 2 x ) = 0 5 x = 0 x = 0 So, ( 0 , 0 ) is the only critical point. Write the system in matrix form Phase plane equation is, d y d x = y − 3 x 2 x + y ( 1 ) Let y = v x . From ( 1 ) , x d v d x = v − 3 2 + v − v x d v d x = − 3 − v 2 − v 2 + v 2 + v − 3 − v 2 − v d v = d x x ∫ 2 + v − 3 − v 2 − v d v = ∫ d x x − 1 2 ln | − v − v 2 − 3 | + 3 2 ∫ d v − v

7th Edition

ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 5.RP, Problem 15RP

To determine

The type of stability of critical points and sketch trajectories.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.RP, Problem 14RP

Chapter 5.RP, Problem 16RP

Students have asked these similar questions

If you are using chatgpt leave it I will downvote .

Temperature measurements are based on the transfer of heat between the sensor of a measuring device (such as an ordinary thermometer or the gasket of a thermocouple) and the medium whose temperature is to be measured. Once the sensor or thermometer is brought into contact with the medium, the sensor quickly receives (or loses, if warmer) heat and reaches thermal equilibrium with the medium. At that point the medium and the sensor are at the same temperature. The time required for thermal equilibrium to be established can vary from a fraction of a second to several minutes. Due to its small size and high conductivity it can be assumed that the sensor is at a uniform temperature at all times, and Newton's cooling law is applicable. Thermocouples are commonly used to measure the temperature of gas streams. The characteristics of the thermocouple junction and the gas stream are such that λ = hA/mc 0.02s-1. Initially, the thermocouple junction is at a temperature Ti and the gas stream at…

A body of mass m at the top of a 100 m high tower is thrown vertically upward with an initial velocity of 10 m/s. Assume that the air resistance FD acting on the body is proportional to the velocity V, so that FD=kV. Taking g = 9.75 m/s2 and k/m = 5 s, determine: a) what height the body will reach at the top of the tower, b) how long it will take the body to touch the ground, and c) the velocity of the body when it touches the ground.

Chapter 5 Solutions

MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The type of stability of critical points and sketch trajectories.

Solution Summary: The author explains the types of stability of critical points and sketch trajectories.

Videos

The type of stability of critical points and sketch trajectories.

Want to see the full answer?

Chapter 5 Solutions