Show that x(t) approaches infinity in finite time

[zooxanthellae]

Homework Statement

Consider the equation $$\dot{x} = rx + x^3,$$ where $$r>0$$ is fixed. Show that $$x(t) \rightarrow \pm \infty$$ in finite time, starting from any initial condition $$x_{0} \neq 0.$$

Homework Equations

I can think of none.

The Attempt at a Solution

The idea alone of x(t) approaching infinity in "finite time" throws me for a loop. Does this merely mean that |x(t)| becomes very large very quickly? That it is increasing (x > 0) or decreasing (x < 0) increasingly quickly as it "moves away from the origin"? It strikes me as impossible for something to "become infinite" in a finite amount of time. This is from Strogatz' "Nonlinear Dynamics and Chaos", Exercise 2.5.3 if you are interested.

Thank you!

 

[hunt_mat]

Solve the equation! You have:
$$\frac{1}{x(r+x^{2})}\frac{dx}{dt}=1$$
Use partial fractions to separate to get:
$$\frac{1}{x(r+x^{2})}=\frac{1}{rx}-\frac{1}{r(r+x^{2})}$$
And then simply integrate, this will give the answer.

 

[Strants]

I would assume the phrase "approaches infinity in finite time" means the graph has an asymptote.

 

[hunt_mat]

Indeed. There is a vertical asymptote as a certain finite value of t.

 

[zooxanthellae]

I think I now understand the math behind this, although I'm unable to come up with a physical situation that models it?

 

[hunt_mat]

First off, can you do the integral?
$$\frac{1}{r}\int \frac{1}{x}-\frac{1}{r+x^{2}}dx=\int 1dt$$
Physically this would represent a shock wave of some kind.

 

[zooxanthellae]

We end up with: $$\frac{ln(x) - (\arctan(x/\sqrt{r})/\sqrt{r})}{4} = t.$$ So if this is a shockwave, what exactly is x(t) modeling? Surely not position?

Sorry for what I can kind of tell are pretty basic questions. I've had very little experience with differential equations so I'm sort of learning as I go along in Strogatz.

 

[hunt_mat]

You also have a constant of integration in there somewhere which you can calculate.

x would represent something like the gradient of the wave at a given time.

So what values on the RHS would make it infinite? look at both the log and arctan terms.

 

[Strants]

Are you sure about that partial fraction simplification? I get
$$\frac{1}{rx} - \frac{x}{r(r+x^2)}$$

 

[hunt_mat]

I think you could be right there...