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Question: symgen order does not match trace information

Question:symgen order does not match trace information

nm 11923 Maple 2025
ode differential-equations symmetry symgen
Yesterday at 2:31 AM
1

Solving this ode, Maple says 

But notice, the xi and eta tangent vectors order is reversed. Maple says pattern is [0,F*G], but shows [F*G,0].

Also, when later using DEtools:-symgen with HINT option, it does not return the above result. Tried both patterns with 0 on left and 0 on right. In both cases symgen does not return what shows above. 

Why is that? Am I doing something wrong? Notice also it says   "way=HINT"  but I am not using way option. Only HINT option, Why is it saying way=HINT? it seems argument passing I  am using is wrong, but do not see why it could be wrong.  This is what help says

Worksheet below

> 

interface(version);

`Standard Worksheet Interface, Maple 2025.2, Windows 10, November 11 2025 Build ID 1971053`

> 

restart;
> 

ode:=diff(y(x),x)*cos(y(x))-cos(x)*sin(y(x))^2-sin(y(x)) = 0;
infolevel[dsolve]:=5;
infolevel[symgen]:=5;

(diff(y(x), x))*cos(y(x))-cos(x)*sin(y(x))^2-sin(y(x)) = 0

5

5

> 

dsolve(ode);

Methods for first order ODEs:

--- Trying classification methods ---

trying a quadrature

trying 1st order linear

trying Bernoulli

trying separable

trying inverse linear

trying homogeneous types:

trying Chini

differential order: 1; looking for linear symmetries

trying exact

Looking for potential symmetries

trying inverse_Riccati

trying an equivalence to an Abel ODE

differential order: 1; trying a linearization to 2nd order

--- trying a change of variables {x -> y(x), y(x) -> x}

differential order: 1; trying a linearization to 2nd order

trying 1st order ODE linearizable_by_differentiation

--- Trying Lie symmetry methods, 1st order ---

 -> Computing symmetries using: way = 3

 -> Computing symmetries using: way = 4

 -> Computing symmetries using: way = 5

trying symmetry patterns for 1st order ODEs

 -> Computing symmetries using: way = patterns

-> trying a symmetry pattern of the form [F(x)*G(y), 0]

-> trying a symmetry pattern of the form [0, F(x)*G(y)]

-> The symmetry found is [0 exp(-x)*(2*sin(y)^2+cos(y)^2-1)/cos(y)]

<- symmetry pattern of the form [0, F(x)*G(y)] successful

[exp(-x)*(2*sin(y)^2+cos(y)^2-1)/cos(y), 0]

 <- successful computation of symmetries.

y(x) = arctan(-2*exp(x)/(exp(x)*cos(x)+exp(x)*sin(x)+2*c__1), ((2*cos(x)*sin(x)*(exp(x))^2+4*c__1*sin(x)*exp(x)+4*cos(x)*c__1*exp(x)+4*c__1^2+(exp(x))^2)*(2*cos(x)*sin(x)*(exp(x))^2+4*c__1*sin(x)*exp(x)+4*cos(x)*c__1*exp(x)-3*(exp(x))^2+4*c__1^2))^(1/2)/(2*cos(x)*sin(x)*(exp(x))^2+4*c__1*sin(x)*exp(x)+4*cos(x)*c__1*exp(x)+4*c__1^2+(exp(x))^2)), y(x) = arctan(-2*exp(x)/(exp(x)*cos(x)+exp(x)*sin(x)+2*c__1), -((2*cos(x)*sin(x)*(exp(x))^2+4*c__1*sin(x)*exp(x)+4*cos(x)*c__1*exp(x)+4*c__1^2+(exp(x))^2)*(2*cos(x)*sin(x)*(exp(x))^2+4*c__1*sin(x)*exp(x)+4*cos(x)*c__1*exp(x)-3*(exp(x))^2+4*c__1^2))^(1/2)/(2*cos(x)*sin(x)*(exp(x))^2+4*c__1*sin(x)*exp(x)+4*cos(x)*c__1*exp(x)+4*c__1^2+(exp(x))^2))

> 

infolevel[dsolve]:=0;

0

> 

DEtools:-symgen(ode,y(x),HINT = [ 0,F(x)*G(y)]); #why this does not return the symmetries found by dsolve above?

 -> Computing symmetries using: way = HINT

> 

DEtools:-symgen(ode,y(x),HINT = [ F(x)*G(y),0]); #why this does not return the symmetries found by dsolve above?

 -> Computing symmetries using: way = HINT

> 

DEtools:-symgen(ode)

 -> Computing symmetries using: way = 2

 -> Computing symmetries using: way = 3

 -> Computing symmetries using: way = 4

 -> Computing symmetries using: way = abaco1

[0, exp(-x)*(-1+cos(2*y))/cos(y)]

 <- successful computation of symmetries.

[_xi = 0, _eta = exp(-x)*(-1+cos(2*y))/cos(y)]

 

 

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