how to check solution?

[Dr Huang]

wolfram give very complicated solution for
y''=y^2-y
how to check its solution?

DrHuang.com

[Axel Vogt]

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]

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]

And what is your solution?

[Dr Huang]

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]

http://server.drhuang.com/input/?guess=dsolve(y(2,x)=y**2-y)

[Axel Vogt]

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]

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]

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]

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]

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)

[Kristjan Robam]

(_!_)