how to check solution?

34 views

Dr Huang

Jun 24, 2022, 1:32:45 AM6/24/22
to
wolfram give very complicated solution for
y''=y^2-y
how to check its solution?

DrHuang.com

Axel Vogt

Jun 26, 2022, 3:19:22 PM6/26/22
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Why should it be 'simple'? I am not into ODE and using Maple it gives
an implicit solution which essentially tells me that one has to invert
Int(1/((6*a^3-9*a^2+9*c)^(1/2)), a = 0 .. z) for z, c is a quadric of
the values for f(0) and f'(0).

In rare cases for c that elliptic integral is 'simple', in general it
is in EllipticF of the (complicated) roots of the cubic. Besides then
solving for z.

The implict solution can be 'comfirmed' using Maple's command 'odetest'.

Dr Huang

Jun 26, 2022, 8:23:22 PM6/26/22
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On Monday, 27 June 2022 at 05:19:22 UTC+10, Axel Vogt wrote:
> Am 24.06.2022 um 07:32 schrieb Dr Huang:
> > wolfram give very complicated solution for
> > y''=y^2-y
> > how to check its solution?
> >
> > DrHuang.com
> Why should it be 'simple'?
Because a simple solution is easy to undersand, plot and check. Why cannot maple find a simple solution? e.g. there are over such 700 ODE in http://drhuang.com/index/bug

Axel Vogt

Jun 27, 2022, 12:41:01 AM6/27/22
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Message has been deleted

Dr Huang

Jun 27, 2022, 11:20:29 PM6/27/22
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Input your ODE into mathHand.com, click the ODE button to solve, or click
http://server.drhuang.com/input/?guess=dsolve%28y%282%2Cx%29%3Dy%5E2-y%29&inp=y%282%2Cx%29%3Dy%5E2-y
then click the test button to test solution

Dr Huang

Jun 27, 2022, 11:30:48 PM6/27/22
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Axel Vogt

Jun 28, 2022, 6:05:47 AM6/28/22
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That link works. So you say it is 1/2+weierstrassP(C_1+x,1/2,C_2)
and no restriction for the constants (?).

What is your convention for weierstrassP ?

anti...@math.uni.wroc.pl

Jun 28, 2022, 7:17:00 AM6/28/22
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In FriCAS:

(22) -> f := 1/2 + 6*weierstrassP(1/12, g2, c1 + x)

1
12 weierstrassP(--,g2,x + c1) + 1
12
(22) ---------------------------------
2
Type: Expression(Integer)
(23) -> D(f, x, 2) - (f^2 - f)

(23) 0
Type: Expression(Integer)

There is also trival solution, that is f := 1. FriCAS definition
of weierstrassP is rather conventional:

(25) -> D(weierstrassP(g2, g3, x), x)

(25) weierstrassPPrime(g2,g3,x)
Type: Expression(Integer)
(26) -> D(weierstrassPPrime(g2,g3,x), x)

2
12 weierstrassP(g2,g3,x) - g2
(26) ------------------------------
2
Type: Expression(Integer)

There seem to be some confusion about order of arguments and
apparently Dr Huang skipped factor of 6 from derivative
of weierstrassPPrime (or whatever he uses instead).

--
Waldek Hebisch

Axel Vogt

Jun 28, 2022, 12:39:08 PM6/28/22
to
Maple (and Mathematica?) uses the following:

diff(f(z), z\$2) = 6*f(z)^2-1/2*g2
has solution
f(z) = WeierstrassP(z+_C1,g2,_C2)

Axel Vogt

Jun 29, 2022, 1:54:31 PM6/29/22
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Am 28.06.2022 um 13:16 schrieb anti...@math.uni.wroc.pl:
> In FriCAS:
>
> (22) -> f := 1/2 + 6*weierstrassP(1/12, g2, c1 + x)
>
> 1
> 12 weierstrassP(--,g2,x + c1) + 1
> 12
> (22) ---------------------------------
> 2
> Type: Expression(Integer)
> (23) -> D(f, x, 2) - (f^2 - f)
>
> (23) 0
> Type: Expression(Integer)
>
> There is also trival solution, that is f := 1. FriCAS definition
> of weierstrassP is rather conventional

Thank you.
In Maple I used: f(x) =1/2 + 6*WeierstrassP(c1 + x, 1/12, c2);
odetest(%, ode) confirming it

Axel Vogt

Jun 30, 2022, 2:08:03 PM6/30/22
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Am 24.06.2022 um 07:32 schrieb Dr Huang:
Somewhat related:

for f'' = f^2 you site gives a rational function 6*(C_1 +- x)^(-2),
http://server.drhuang.com/input/?guess=dsolve(y(2,x)=y**2),
only one parameter

However Maple gives 6*WeierstrassP(z+_C1,0,_C2) for which your output
is a special and limiting case (discriminant = 0)