Explanation Given information: Given differential equation is d y d t = 1 + cos y Concept Used: To draw the phase line of an autonomous equation d y d t = f ( y ) , we follow the steps: First we find the equilibrium points f ( y ) = 0 and mark them on the y -axis Now let’s find the intervals for which f ( y ) > 0 , and draw arrows pointing upward in these intervals. Find the intervals of y -values for which f ( y ) < 0 , and draw arrows pointing downward in these intervals. For an equilibrium point of the differential equation , d y d t = f ( y ) , a if f ′ ( a ) < 0 , then the point a is sink if f ′ ( a ) > 0 , then the point a is source if f ′ ( a ) = 0 , then the point a is node Calculation: To draw the phase line of the differential equation d y d t = 1 + cos y First, we need to find the equilibrium point by putting d y d t = f ( y ) = 0 , as 1 + cos y = 0 ⇒ cos y = − 1 ⇒ y = ( 2 n + 1 ) π Where n = 0 , ± 1 , ± 2 , ± 3..... Thus, the equilibrium points are ..... , − 3 π , − π , π , 3 π ..... Now, we have to check the signs of f ( y ) in the intervals ....... , ( − 3 π , − π ) , ( − π , π ) , ( π , 3 π ) , ...

4th Edition

ISBN: 9780495561989

Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 1.6, Problem 9E

To determine

To draw the phase line for the given differential equation. And to identify the equilibrium points as sinks, sources, or nodes

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 1.6, Problem 8E

Chapter 1.6, Problem 10E

Students have asked these similar questions

5. (a) State the Residue Theorem. Your answer should include all the conditions required for the theorem to hold. (4 marks) (b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the anti-clockwise direction. Evaluate に dz. You must check all of the conditions of any results that you use. (5 marks) (c) Evaluate L You must check all of the conditions of any results that you use. ཙ x sin(Tx) x²+2x+5 da. (11 marks)

3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula for L(y). (1 mark) (b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a contour. Suppose there exists a finite real number M such that |f(z)| < M for all z in the image of y. Prove that < ||, f(z)dz| ≤ ML(y). (3 marks) (c) State and prove Liouville's theorem. You may use Cauchy's integral formula without proof. (d) Let R0. Let w € C. Let (10 marks) U = { z Є C : | z − w| < R} . Let f UC be a holomorphic function such that 0 < |ƒ(w)| < |f(z)| for all z Є U. Show, using the local maximum modulus principle, that f is constant. (6 marks)

3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M a simple module? (b) State and prove Schur's Lemma for simple modules. (c) Let AM(K) and M = K" the natural A-module. (i) Show that M is a simple K-module. (ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a is a matrix in the centre of M, (K). [Recall that the centre, Z(M,(K)) == {a Mn(K) | ab M,,(K)}.] = ba for all bЄ (iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~ K as K-algebras. Is this consistent with Schur's lemma?

Chapter 1 Solutions

Differential Equations

Ch. 1.1 - In Exercises 1 and 2, find the equilibrium...Ch. 1.1 - In Exercises 1 and 2, find the equilibrium...Ch. 1.1 - Consider the population model dPdt=0.4P(1P230)...Ch. 1.1 - Consider the population model ...Ch. 1.1 - Consider the differential equation dydt=y3y212y...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...Ch. 1.1 - In Exercises 6—10, we consider the phenomenon of...

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.

To draw the phase line for the given differential equation. And to identify the equilibrium points as sinks, sources, or nodes

Solution Summary: The author explains how to draw the phase line for a given differential equation and identify the equilibrium points as sinks, sources, or nodes.

Concept explainers

Videos

To draw the phase line for the given differential equation. And to identify the equilibrium points as sinks, sources, or nodes

Want to see the full answer?

Chapter 1 Solutions