2nd order DEQ, conserved quantity

[fusi0n]

Homework Statement

Given: y'' - y - (y^3) = 0 (equation 1)

E = (1/2)(v^2) - (1/2)(y^2) - (1/4)(y^4) (equation 2)

v = y'

i. Show that E is a conserved quanitity
ii. Find all the solutions with E = 0

2. The attempt at a solution

I'm not sure how to show a quantity is being conserved. In fact, I have no idea how to begin this problem! Does anybody have some information to help me just get started?

I solved equation 2 for v = (E + y^2 + (1/2)y^4)^(1/2). I realize I can integrate and solve for y(t) but it is really very messy and I don't see how doing so can immediately benefit me.

 

[HallsofIvy]

"Conserved" (as in conservation of energy and conservation of momentum) means "does not change". I.e. the derivative is 0.

To show that E is a "conserved quantity", differentiate E (with respect to whatever the independent variable is) and use the given differential equation (y"= y+ y³) to show that the derivative is 0.

 

[fusi0n]

Question: How do I differentiate E with respect to t when y and v are dependent upon t?

here is my attempt...

E = (1/2)(v^2) - (1/2)(y^2) - (1/4)(y^4) where v = (dy/dt) =>

dE/dt = (dv/dt) - (dy/dt) - (dy/dt) = y'' - 2y'

 

[cristo]

Question: How do I differentiate E with respect to t when y and v are dependent upon t?

Use the chain rule. For example, $$\frac{d}{dt}(y^2)=\frac{d}{dy}(y^2)\frac{dy}{dt}$$

 

[fusi0n]

thank you everyone; I have proven that E is conserved.