
Life lessons from differential equations (2015) - magoghm
https://www.johndcook.com/blog/2015/07/23/life-lessons-from-differential-equations/
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chrbarrol
I felt that one addition could be "Many problems have trivial solutions, but
they are of little practical value"

0 is often a trivial solution to many PDEs, but it is of little analytical
value, so it is often discarded. In the same sense death is a trivial of many
life problems: Dying would solve most (all?) problems in life, but it is not a
solution you would typically consider :)

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stabbles
It all depends on the boundary conditions.

~~~
goldenkey
My boundary condition is fulfilling the categorical imperative of pure energy.
For some people its the imperative of themselves or immediate families, or
mammals or life itself. I still haven't figured out what the universe would
like but I am at its service.

My best guess at the current moment is to extend the universe's life but the
inevitable heat death if correct throws a wrench in that.

However if there is a way to create continuity from discreteness, and we can
simulate our universe, we may be able to run a child universe to completion
before our own universe dies. And if the same thing happens in that child, we
will have had an infinite number of universes live and die in the finite life
of our own universe. If that isn't a full life for a universe, I don't know
what is...

This already may be happening if black holes birth and contain universes.

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smaddox
> I still haven't figured out what the universe would like but I am at its
> service.

If you're trying to figure out "what the universe would like", you're going to
be searching until the heat death of the universe.

> However if there is a way to create continuity from discreteness, and we can
> simulate our universe, we may be able to run a child universe to completion
> before our own universe dies.

This is ontologically unworkable. The sooner you accept that nothing will last
forever, the sooner you can get to living your life and enjoying the things
that are here right now.

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Yhippa
One of the most important lessons I learned was spending several pages of
notebook paper devising a clever solution only to find that it completely
doesn't work. Pretty darn good way to humble yourself.

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ianai
It’s important to try many different approaches to difficult problems. Or, try
different problems with new approaches - to learn about the approaches.

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esturk
One of the points link to a second article that reads:

(The fact that φ is zero outside a finite interval mean the “uv” term from
integration by parts is zero.)

Can anyone elaborate on this? I'm not too sure how this trick works.

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pedrosorio
This is the article you're referring to:
[https://www.johndcook.com/blog/2009/10/25/how-to-
differentia...](https://www.johndcook.com/blog/2009/10/25/how-to-
differentiate-a-non-differentiable-function/)

This is the wikipedia page on integration by parts:
[https://en.wikipedia.org/wiki/Integration_by_parts](https://en.wikipedia.org/wiki/Integration_by_parts)

He's setting u = phi(x), v = f(x).

Since he's integrating on the real line, a = -infinity, b = +infinity, so on
the right hand side for "uv" you'd have u(+inf) _v(+inf) - u(-inf)_ v(-inf) -
and since u (which is phi) is 0 outside of a finite interval, you know these
terms are 0.

[You obviously can't evaluate functions at +-inf, and you have to take limits
to evaluate u(x)v(x) for an improper integral, but you can see the result is
the same if u(x) is zero outside of a finite interval]

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esturk
Oh I see. So in the evaluation of u(+inf), it becomes 0, hence uv is 0. I have
never thought of it that way. Thanks!

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paulpauper
boundary conditions do indeed make solving differential equations much harder

thankfully the use of infinite series can help a lot in such situations

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garmaine
> Some problems simply have no solution

Don't confuse "exact solution" with "solution" (unqualified). There is no
problem that has no solution.

I stopped reading there.

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prmph
What is the solution for avoiding death and taxes?

~~~
severino
Dying is not always a problem. In fact, lots of people consider it a solution.

The same can be said about taxes. They were the solution for problems like
giving people access to education and healthcare.

