ODE: Infinite Solutions with Initial Condition x(0) = 0

[Niles]

Homework Statement

Hi all.

I am given the following ODE:

$$\frac{dx}{dt} = x^{1/3}$$

with the initial condition x(0) = 0. I have to show that there is an infinite number of solutions. I'm not quite sure how to get started other than using separation of variables and going on from there. But this only gives me 1 solution other than the trivial solution x(t) = 0.

So I have 2 solutions, which is less than infinite. What approach should I use?

Thanks in advance.


Niles.

 

[Dick]

Here's another solution. Let x(t)=0 on [0,1]. Now solve your ODE on the region [1,infinity) by solving the equation with the initial condition x(1)=0 and pick the nontrivial solution. I.e. define x(t) piecewise instead of just as a single formula. There's lots of solutions like this, right?

 

[Niles]

Thanks. But doesn't this imply that each ODE has infinitely many solutions? I mean, our tactic in this case is just to make the solution piecewise, where the first piece is just 0. Couldn't something similar be done to an arbitrary ODE?

By the way, am I allowed to ask what subfield of physics you did your primary research in?

 

[Dick]

Thanks. But doesn't this imply that each ODE has infinitely many solutions? I mean, our tactic in this case is just to make the solution piecewise, where the first piece is just 0. Couldn't something similar be done to an arbitrary ODE?

By the way, am I allowed to ask what subfield of physics you did your primary research in?

No, you can't do this trick with any ODE. Try it with a regular ODE. When you specify the initial conditions for the next piece in such a way as to make it continuous with the last piece, you get the same solution. It works for this one because there are two different solutions corresponding to the same initial conditions. Sure you can ask. I did my graduate work in cosmology.

 

[Niles]

Thanks for explaining that. I just need clarification of one single thing: When you say "solve the ODE on the interval [0;1]", you are just telling me to solve the following ODE with the initial condition x(1)=0, right?

$$\frac{dx}{dt} = x^{1/3}.$$

Was there a particular reason why you chose cosmology, or where you just following your interest?

 

[Dick]

Sure, just splice the zero solution on [0,1] to the nonzero solution on [1,infinity). Just following my interests, in the cosmology thing. Always had a thing for GR.

 

[Niles]

Great, thanks for that. You seem very talented, so the award is well deserved.