Take any differential equation of the form
$$\frac{dy}{dx}=y^n$$
where $n > 1$. The solution $y(x)$ will reach infinity at a finite value of $x$.
Assuming $y_0 =1 $ for all cases, here are a few examples:
$$\frac{dy}{dx}=y^2$$ has the solution $$y=\frac{-1}{x-1}$$
which reaches its asymptote at $x=1$.
The DE
$$\frac{dy}{dx}=y^{1.01}$$ has the solution $$y=\left(\frac{-100}{x-100}\right)^{100}$$
which reaches its asymptote at $x=100$.
If you take any DE of the form
$$\frac{dy}{dx}=y^{1 + \epsilon}$$
where $\epsilon$ is a very small number, the solution is
$$y=\left(\frac{-1}{\epsilon(x-\frac{1}{\epsilon})}\right)^{\epsilon^{-1}}$$
which eventually hits the vertical asymptote at the very large number $\frac{1}{\epsilon}$
This has always bugged me. Intuitively, one expects that the solutions to these equations will grow rapidly and aggressively, much faster than the exponential function. But it is not entirely obvious why they should reach an infinite value after a finite time, instead of say, grow like the Ackermann function or some other function that grows rapidly but stays strictly finite.
Is there an intuitive argument for why these DEs are able to reach infinity in a finite timespan?
Best Answer
The point is that $dy/dx = y^p$ is equivalent to $dx/dy = y^{-p}$, i.e. instead of thinking of $y$ as the dependent variable and $x$ as independent, do the reverse. If you think of $x$ as position and $y$ as time, the velocity is $y^{-p}$. If $p > 1$, this goes to $0$ fast enough that the change in $x$ as $y$ goes from some finite positive value to $\infty$ is finite. Now change point of view again and it says that as $x$ goes to some finite value, $y$ goes to $\infty$.