Discussion:
Does this differential equation have a solution?
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Robert Delaney
2015-05-22 01:09:59 UTC
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For the differential equation:

dy/dx = - sqrt(y - 1/x)
 
can a real solution y(x) be found for which:
 
lim(x->inf) y(x) = 0
 
Squaring both sides of the equation, I have found the "solution":

y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 +
3386880/x^19 + 146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 + 1641030725533696/x^34 + 119809030723207168/x^37 +


which appears to be an asymptotic power series. From Wikipedia
"Typically, the best approximation is given when the series is
truncated at the smallest term." So as x->inf it does appear that y->0.
But I have not found a convergent power series solution nor any closed
form solution of the type described. Does one exist?

Bob
William Elliot
2015-05-22 03:17:49 UTC
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Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 + 3386880/x^19 +
146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 + 24450285436928/x^31 +
1641030725533696/x^34 + 119809030723207168/x^37 + 

which appears to be an asymptotic power series. From Wikipedia "Typically, the
best approximation is given when the series is truncated at the smallest
term." So as x->inf it does appear that y->0. But I have not found a
convergent power series solution nor any closed form solution of the type
described. Does one exist?
Bob
Thomas
2015-05-22 12:53:09 UTC
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I can't find this thread in sci.math. Where else is it posted?
Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
(*) From what my ode-solver suggests it seems that all solutions of the
diff. eq. with y(x0) = y0, x0>0, y(x0) > 1/x0 have this property.
Post by Robert Delaney
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 + 3386880/x^19 +
146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 + 24450285436928/x^31 +
1641030725533696/x^34 + 119809030723207168/x^37 + …
I hand-checked this up to the 1/x^10 term. Seems fine. But I don't see
how this helps to find any particular solution or to prove anything.
Post by Robert Delaney
which appears to be an asymptotic power series. From Wikipedia "Typically, the
best approximation is given when the series is truncated at the smallest
term." So as x->inf it does appear that y->0. But I have not found a
convergent power series solution nor any closed form solution of the type
described. Does one exist?
A power-series expansion will only converge on finite intervals. I may
give it a try. But then, what does it tell us? But maybe one can prove
my claim (*) in a direct way without computing any expansions.
Post by Robert Delaney
Bob
--
Thomas
Bill
2015-05-22 12:55:59 UTC
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Post by Thomas
I can't find this thread in sci.math. Where else is it posted?
Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
(*) From what my ode-solver suggests it seems that all solutions of
the diff. eq. with y(x0) = y0, x0>0, y(x0) > 1/x0 have this property.
Post by Robert Delaney
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 + 3386880/x^19 +
146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 +
1641030725533696/x^34 + 119809030723207168/x^37 + …
I hand-checked this up to the 1/x^10 term. Seems fine. But I don't see
how this helps to find any particular solution or to prove anything.
Post by Robert Delaney
which appears to be an asymptotic power series. From Wikipedia "Typically, the
best approximation is given when the series is truncated at the smallest
term." So as x->inf it does appear that y->0. But I have not found a
convergent power series solution nor any closed form solution of the type
described. Does one exist?
A power-series expansion will only converge on finite intervals.
This is not necessarily true. Look at the one for e^x.
Post by Thomas
I may give it a try. But then, what does it tell us? But maybe one can
prove my claim (*) in a direct way without computing any expansions.
Post by Robert Delaney
Bob
Thomas
2015-05-22 13:35:55 UTC
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Post by Bill
Post by Thomas
I can't find this thread in sci.math. Where else is it posted?
Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
(*) From what my ode-solver suggests it seems that all solutions of
the diff. eq. with y(x0) = y0, x0>0, y(x0) > 1/x0 have this property.
Post by Robert Delaney
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 + 3386880/x^19 +
146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 +
1641030725533696/x^34 + 119809030723207168/x^37 + …
I hand-checked this up to the 1/x^10 term. Seems fine. But I don't see
how this helps to find any particular solution or to prove anything.
Post by Robert Delaney
which appears to be an asymptotic power series. From Wikipedia "Typically, the
best approximation is given when the series is truncated at the smallest
term." So as x->inf it does appear that y->0. But I have not found a
convergent power series solution nor any closed form solution of the type
described. Does one exist?
A power-series expansion will only converge on finite intervals.
This is not necessarily true. Look at the one for e^x.
Well, of course. Thanks.
--
Thomas
Thomas
2015-05-22 14:06:00 UTC
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Post by Thomas
I can't find this thread in sci.math. Where else is it posted?
Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
(*) From what my ode-solver suggests it seems that all solutions of the
diff. eq. with y(x0) = y0, x0>0, y(x0) > 1/x0 have this property.
Post by Robert Delaney
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 + 3386880/x^19 +
146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 +
1641030725533696/x^34 + 119809030723207168/x^37 + …
I hand-checked this up to the 1/x^10 term. Seems fine. But I don't see
how this helps to find any particular solution or to prove anything.
I must correct myself. If one could show that this _is_ an asymptotic
expansion, that is y(x) = 1/x + O(1/x) as x->oo (we really do not need
any more terms) it follows that y(x) -> 0. But right now I'm not sure
whether equating coefficients will do the job.
--
Thomas
Robert Delaney
2015-05-22 21:10:52 UTC
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Post by Thomas
Post by Thomas
I can't find this thread in sci.math. Where else is it posted?
Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
(*) From what my ode-solver suggests it seems that all solutions of the
diff. eq. with y(x0) = y0, x0>0, y(x0) > 1/x0 have this property.
Post by Robert Delaney
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 + 3386880/x^19 +
146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 +
1641030725533696/x^34 + 119809030723207168/x^37 + …
I hand-checked this up to the 1/x^10 term. Seems fine. But I don't see
how this helps to find any particular solution or to prove anything.
I must correct myself. If one could show that this _is_ an asymptotic
expansion, that is y(x) = 1/x + O(1/x) as x->oo (we really do not need
any more terms) it follows that y(x) -> 0. But right now I'm not sure
whether equating coefficients will do the job.
The coefficients can be generated from:

c(1) = 1
c(3k+1) = sum(n=0 to k-1) (3n+1)[3(k-n)-2]c(3n+1)c(3(k-n)-2) ; k = 1, 2, 3, …

The needed values of the coefficients on the rhs have always been
previously calculated.

Perhaps one can use this for proving the power series is asymptotic.

Bob
Thomas
2015-05-22 14:58:23 UTC
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Post by Thomas
I can't find this thread in sci.math. Where else is it posted?
Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
lim(x->inf) y(x) = 0
(*) From what my ode-solver suggests it seems that all solutions of the
diff. eq. with y(x0) = y0, x0>0, y(x0) > 1/x0 have this property.
...
Post by Thomas
[...] But maybe one can prove
my claim (*) in a direct way without computing any expansions.
OK, maybe this works:

Let x0 > 0, y0 > 1/x0. Let y(x) be the solution of the diff. eqn. with
these initial conditions. Since dy/dx = - sqrt(y - 1/x) <=0 y(x) is
decreasing for all x in its interval of existence. There are three cases:

1) y(x1) = 1/x1 at some x1 > x0 (Right end point of existence-interval)
and y(x) > 1/x for x < x1.
2) y(x) -> a>0 for some a>0 as x -> oo
3) y(x) -> 0 as x -> oo

Therefore I want to show that 1) and 2) are not possible:
Proof. 2): In that case dy/dx -> -sqrt(a) < 0 as x -> oo. This is not
possible because then y(x) -> - oo as x -> oo

1). Let u = y - 1/x then the diff. eq. transforms to

du/dx = 1/t^2 - sqrt(u), u(x0) = u0 > 0.

If y(x1) = 1/x1 then u(x1) = 0 and du/dx = 1/(x1)^2 > 0 at x=x1. But
then u(x) < 0 for x<x1, x sufficiently close to x1. This is not possible
by assumption. Therefore only case 3) remains.
--
Thomas
d***@gmail.com
2015-05-22 04:44:44 UTC
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The RHS of the ode becomes imaginary / undefined if y drops to 0 as x increases beyond bounds, so dy/dx cannot be defined. Doubtful if we will find a nice y(x).
Robert Delaney
2015-05-22 05:42:25 UTC
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Post by d***@gmail.com
The RHS of the ode becomes imaginary / undefined if y drops to 0 as x
increases beyond bounds, so dy/dx cannot be defined. Doubtful if we
will find a nice y(x).
Unless y drops to zero always more slowly than does 1/x. Then the RHS
never goes imaginary during the approach to the limit.

Bob
Robert Delaney
2015-05-24 01:15:38 UTC
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Post by Robert Delaney
dy/dx = - sqrt(y - 1/x)
 
 
lim(x->inf) y(x) = 0
 
y = 1/x + 1/x^4 + 8/x^7 + 128/x^10 + 3008/x^13 + 91584/x^16 +
3386880/x^19 + 146442240/x^22 + 7222489088/x^25 + 399385972736/x^28 +
24450285436928/x^31 + 1641030725533696/x^34 + 119809030723207168/x^37 +

which appears to be an asymptotic power series. From Wikipedia
"Typically, the best approximation is given when the series is
truncated at the smallest term." So as x->inf it does appear that y->0.
But I have not found a convergent power series solution nor any closed
form solution of the type described. Does one exist?
Bob
A numerical solution of the differential equation shows an
insensitivity to the initial value of y for a given initial value for
x. And the numerical solution tracks closer and closer to the leading
terms of the asymptotic power series.

For example, using Runge-Kutta with x-spacing=0.001

startx = 2.0
starty = 0.6
one finds when
x = 10
y = 0.100100813110366

and for

startx = 2.0
starty = 0.66
one finds when
x = 10
y = 0.100100813110366

showing the same final values for y to about 15 decimal places for a
10% difference in the starty values.

Summing the asymptotic power series to and including the 1/x^25 term
gives for x = 10:

y = 0.100100813110312

which differs only in the final two digits from the Runge-Kutta value.

Bob

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