# 2nd order DEQ, conserved quantity

• fusi0n
In summary, the conversation discusses an equation (equation 1) and a conserved quantity (equation 2) involving y'' and y. The task is to show that E is a conserved quantity and to find all solutions for E=0. The attempt at a solution involves solving equation 2 for v and using the chain rule to differentiate E with respect to t. The conversation ends with the acknowledgement that E is indeed conserved.

## 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.

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"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.

Question: How do I differentiate E with respect to t when y and v are dependant 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'

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fusi0n said:
Question: How do I differentiate E with respect to t when y and v are dependant upon t?

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

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

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