Solution of a differential equation having a singularity (not everywhere defined)

Question

Remind me about ordinary differential equations, whose solutions are not everywhere defined (have a singularity).

I want to remember the exact definition of a solution with singularity, which I studied in a university long time ago.

And please, an example.

Answer 1

A solution $x(t)$ of the ODE is called maximal if it is defined on an open interval and cannot be extended to any larger open interval.

From "Ordinary Differential Equation". Alexander Grigorian. University of Bielefeld. Lecture Notes, April - July 2008

So the things I search for are maximal solutions.

The question remains: What is the best generalization of this for PDE?

Answer 2

The easiest example is the `finite time blow-up' ODE:

$$ y^\prime =y^2,\qquad y(0)=1 $$

This equation has the unique solution

$$ y(t)=\frac{1}{1-t} $$ which is not defined when $t=1$. So technically speaking, $y(t)$ isn't a solution to $y^\prime=y^2$ for all time - we have to choose an interval where the solution makes sense. The rigorous approach would go as follows:

We say that $y(t)$ is a (classical) solution to the following ODE

$$ y^\prime=f(t,y),\qquad y(0)=y_0,\qquad a<t<b $$ If

1. $y^\prime(t)$ exists for all $a<t<b$ (note this requires that $y(t)$ itself exists for all $a<t<b$)
2. $y^\prime(t)=f(t,y)$ for all $a<t<b$
3. $y(0)=y_0$

Any solution which fails to satisfy these conditions at some point could be called a `singular' solution. The example $y^\prime=y^2$ violates the first condition: $y^\prime$ doesn't exist when $t=1$. However, if we restrict the problem to read as follows:

$$ y^\prime=y^2,\qquad y(0)=1,\qquad 0\leq t<1 $$ Then $y(t)=(1-t)^{-1}$ is a classical solution. Note that I have included the left endpoint assuming the appropriate definition of the derivative at $t=0$.

There is much more - I recommend picking up a book on ODE theory e.g. Hartman or Arnold.

Answer 3

Judging from your comment below icurays1's answer, I think you're simply interested in solving differential equations on disconnected domains.

You might take some differential equation and try to find a solution on $\mathbb R\setminus D$, where $D$ is a discrete set of points. (This would rule out the empty function, as $D$, being discrete, cannot be all of $\mathbb R$.) For what it formally means to be a solution, see icurays1's answer: basically the appropriate derivatives must exist at all points of $\mathbb R\setminus D$ and the equation must be satisfied at each point of $\mathbb R\setminus D$.

For example, the functions $f:\mathbb R\setminus\{0\}\to\mathbb R$ satisfying the equation $f'=0$ are precisely the functions which are equal to some constant $C_1$ on $(-\infty,0)$ and equal to some (possibly different) constant on $C_2$ on $(0,\infty)$. More generally, I think you might also be interested in the concept of a locally constant function.