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.