Critical points of differential equation

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Homework Help Overview

The discussion revolves around determining the critical points of the second-order differential equation y'' + cos(y) = 0 and sketching the corresponding phase portrait. Participants are exploring the nature of these critical points and their classification.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss converting the second-order equation into a system of first-order equations. There is uncertainty about the classification of critical points, with questions about maxima and minima arising. Some participants suggest setting derivatives to zero to find critical points.

Discussion Status

The discussion is active, with participants sharing their attempts to reformulate the problem and identify critical points. Guidance has been offered regarding the conversion of the equation into a system, and there is an ongoing exploration of the nature of the critical points identified.

Contextual Notes

Participants are navigating the complexities of second-order differential equations and the implications of their findings on the classification of critical points. There is mention of using external resources for visualizing the phase portrait.

samee
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Homework Statement



Determine the location and type of all critical points of the given equations and sketch the phase portrait

y"+cosy=0



The Attempt at a Solution



I've done some like this before but they were all systems of equations. I'm actually not sure how to do the simpler version >_< I think this is asking for maxima and minima? is that right?
 
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It's NOT simpler. This is only one equation but is second order. What you want is two first order equations. Define x by x= y'. Then y''= x' so y''+ y= 0 becomes x'+ y= 0 or x'= -y.

You now have the two equations x'= -y and y'= x.
 
Okay, so for my problem I have
x=y'
x'=y"
so I substitute and
x'+cosy=0
x'=-cosy

so my system of equations is;

y'=x
x'=-cosy

right? Then I just solve like it's a system of equations and look for the singularities as the critical points?
 
Ah! No wait, there's more! I know what I'm doing now, silly me. I set x' and y' to zero and solve for the points.

x'=0, ∴x=0
y'=0, -cosy=0, ∴y=(1/2)(2n+1)pi

So I have infinite points along the y-axis. I used

http://www.math.rutgers.edu/courses/ODE/sherod/phase-local.html

to graph the phase portrait and found that every other critical point is a saddle point, then a center point.

SO! how do I say that one is a saddle, then the next is a center?
 
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