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Chapter 5: Q22E (page 271)

In Problems 19鈥�24, convert the given second-order equation into a first-order system by setting v=y鈥�. 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).

d2ydt2+y3=0.

Short Answer

Expert verified

The point is the center point (0, 0).

Step by step solution

01

Find the critical point

Here the equation is d2ydt2+y3=0.

Put v=y'鈥�鈥�and鈥�鈥�v'=y''.

Then the system is;

y'=vy''=-y3v'=-y3

For critical points equate the system equal to zero.

v=0-y3=0y=0

So, the critical point is (0, 0).

The phase plane equation is;

dvdy=-yv鈭�vdv=鈭�-y3dyv22=-y44+c4v2+y2=c

02

Sketch

Therefore, thepoint is the center point (0, 0).

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Most popular questions from this chapter

In Problems 11鈥�14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

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In Problems 19鈥�24, convert the given second-order equation into a first-order system by setting v=y鈥�. 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).

d2ydt2+y=0

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