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Consider the system of differential equations in the image where V(x)=x^2+εx^4   with ε≥0: 1. Solve the system if ε = 0. 2.Determine all rest points 3.Proof that (dH)/(dt)=0 with H= (y^2)/2+V(x) 4.Prove that every orbit that does not pass through (0, 0) is periodic.

Consider the system of differential equations in the image where V(x)=x^2+εx^4   with ε≥0: 1. Solve the system if ε = 0. 2.Determine all rest points 3.Proof that (dH)/(dt)=0 with H= (y^2)/2+V(x) 4.Prove that every orbit that does not pass through (0, 0) is periodic.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the system of differential equations in the image where V(x)=x^2+εx^4   with ε≥0:

1. Solve the system if ε = 0.

2.Determine all rest points

3.Proof that (dH)/(dt)=0 with H= (y^2)/2+V(x)

4.Prove that every orbit that does not pass through (0, 0) is periodic.

 

If you could provide me with some explanation with the taken steps I would be very thankful, thanks in advance.

 

 

 

 

x ý = = Y, -V'(x)
Transcribed Image Text:x ý = = Y, -V'(x)
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