Answer: In 19–24, convert the given second-order equation

Chapter 5, Problem 20E

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QUESTION:

In Problems 19-24, convert the given second-order equation into a first-order system by setting \(v=y^{\prime}\) Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).
\(\frac{d^{2} y}{d t^{2}}+y=0\)

Equation transcription:

Text transcription:

v=y^{prime}

frac{d^{2} y}{d t^{2}}+y=0

Questions & Answers

QUESTION:

In Problems 19-24, convert the given second-order equation into a first-order system by setting \(v=y^{\prime}\) Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).
\(\frac{d^{2} y}{d t^{2}}+y=0\)

Equation transcription:

Text transcription:

v=y^{prime}

frac{d^{2} y}{d t^{2}}+y=0

ANSWER:

Solution :

Step 1 :

In this problem we have to find all the critical points in the plane and describe the stability of the critical points .

Given the differential equation is

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