Time to Travel to One Infinity and Back?

[royblaze]

Homework Statement

Consider the ODE x' = x² + ε, where ε is a small number. Find the time T = T(ε) it takes for the solution to travel from x = -∞ to x = ∞.

Let T₁ = T₁(ε) be the time it takes the particle to travel to x = -1 to x = 1. Show that T/T₁ → 1 as ε → 1.

Homework Equations

Uhmm... we haven't really learned any yet. This is qualitative analysis of ODEs.

The Attempt at a Solution

I have no idea where to start really. I thought that to find time, we could treat x' as a velocity, and find the position function by integrating x'. Once we had this position function, I thought it would be okay to divide position by velocity to find time... but I don't even know what that would represent! From negative infinity to infinity, no less...

 

[Dick]

Homework Statement

Consider the ODE x' = x² + ε, where ε is a small number. Find the time T = T(ε) it takes for the solution to travel from x = -∞ to x = ∞.

Let T₁ = T₁(ε) be the time it takes the particle to travel to x = -1 to x = 1. Show that T/T₁ → 1 as ε → 1.

Homework Equations

Uhmm... we haven't really learned any yet. This is qualitative analysis of ODEs.

The Attempt at a Solution

I have no idea where to start really. I thought that to find time, we could treat x' as a velocity, and find the position function by integrating x'. Once we had this position function, I thought it would be okay to divide position by velocity to find time... but I don't even know what that would represent! From negative infinity to infinity, no less...

In this case you have to solve the ODE before you start making statements about the qualitative behavior. Start with the second part. Solve the ODE with initial conditions x=(-1) at t=0 and figure out how long it takes you to get to x=1.

 

[royblaze]

Are the initial conditions arbitrarily set? As in it's just convenient to use -1 at t=0 because... well I don't know... :/

I mean, how exactly did you come to the conclusion that that was the best course of action to start with?

 

[Dick]

Are the initial conditions arbitrarily set? As in it's just convenient to use -1 at t=0 because... well I don't know... :/

I mean, how exactly did you come to the conclusion that that was the best course of action to start with?

I think it's the best course of action because I don't see how else you would do it. And no, the initial conditions aren't particularly arbitrary. If x(0)=(-1) then if you can find the value of t such that x(t)=1 (or at least the limit of that as epsilon goes to 0) then that value of t is the time required to go from x=(-1) to x=1.

 

[royblaze]

So I am trying to solve it now, but I am a little put off by the ε. I mean, I understand how to solve an ODE with just x' = x²... but when you say the limit of ε → 0, is that a final step or something? I'm just having trouble getting this problem off the ground.

EDIT: I'd like to add that my professor "kind of" went over this question today. He stated that if ε > 0, then there would be no equilibrium solutions, but if ε < 0, then there are two equilibrium solutions on a "bifurcation line" thing. I remember the term "saddle node bifurcation" and that if ε < 0, then the two equilibrium solutions are -√ε and √ε. I feel that I should use this information to solve this question... but to be honest I didn't really understand any of this bifurcation stuff.

 

[Dick]

So I am trying to solve it now, but I am a little put off by the ε. I mean, I understand how to solve an ODE with just x' = x²... but when you say the limit of ε → 0, is that a final step or something? I'm just having trouble getting this problem off the ground.

EDIT: I'd like to add that my professor "kind of" went over this question today. He stated that if ε > 0, then there would be no equilibrium solutions, but if ε < 0, then there are two equilibrium solutions on a "bifurcation line" thing. I remember the term "saddle node bifurcation" and that if ε < 0, then the two equilibrium solutions are -√ε and √ε. I feel that I should use this information to solve this question... but to be honest I didn't really understand any of this bifurcation stuff.

You really need to be able to solve differential equations to understand them. Forget all the other stuff for now. Just concentrate on that. Put ε=1. Try and solve x'=x^2+1.