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Question: odeadvisor, possible wrong classification on first order ode

Question:odeadvisor, possible wrong classification on first order ode

nm 9899 Maple 2024
ode dsolve differential-equations odeadvisor
June 18 2024
1

odeadvisor says that this ode is _homogeneous, `class A`, but I am not able to verify this. Also when asking dsolve to solve it as 'homogeneous' it returns no solution. 

This type is described in https://www.maplesoft.com/support/help/maple/view.aspx?path=odeadvisor%2fhomogeneous

Here is worksheet with my tries.

Would someone be able to confirm if this is really an _homogeneous, `class A` ?

my own code checking says no.  But if it is, then why dsolve do not solve it when asking it to use homogeneous method? Is the method I asked it to use it do not apply to class A?

30348

> 

restart;

30348

> 

ode:=x + diff(y(x), x)*y(x)*(2*diff(y(x), x)^2 + 3) = 0;
DEtools:-odeadvisor(ode);

x+(diff(y(x), x))*y(x)*(2*(diff(y(x), x))^2+3) = 0

[[_homogeneous, `class A`], _dAlembert]

> 

infolevel[dsolve]:=5;
dsolve(ode,y(x))
> 

 

5

Methods for first order ODEs:

   *** Sublevel 2 ***

   Methods for first order ODEs:

   -> Solving 1st order ODE of high degree, 1st attempt

   trying 1st order WeierstrassP solution for high degree ODE

   trying 1st order WeierstrassPPrime solution for high degree ODE

   trying 1st order JacobiSN solution for high degree ODE

   trying 1st order ODE linearizable_by_differentiation

   trying differential order: 1; missing variables

   trying simple symmetries for implicit equations

   <- symmetries for implicit equations successful

y(x) = -((1/2)*I)*2^(1/2)*x, y(x) = ((1/2)*I)*2^(1/2)*x, y(x) = RootOf(-ln(x)+Intat(-(-2*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*_a^2+2*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)*_a^3-((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)+_a*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)+_a^2)/(((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)*(2*_a^4+3*_a^2+1)), _a = _Z)+c__1)*x, y(x) = RootOf(-2*ln(x)+Intat(((2*I)*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*3^(1/2)*_a^2+I*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*3^(1/2)-2*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*_a^2-4*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)*_a^3+I*3^(1/2)*_a^2-((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)-2*_a*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)+_a^2)/(((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)*(2*_a^4+3*_a^2+1)), _a = _Z)+2*c__1)*x, y(x) = RootOf(-2*ln(x)-Intat(((2*I)*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*3^(1/2)*_a^2+I*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*3^(1/2)+I*3^(1/2)*_a^2+2*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)*_a^2+4*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)*_a^3+((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(2/3)+2*_a*((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)-_a^2)/(((_a^2-(2*_a^2+1)^(1/2)+1)*_a/(2*_a^2+1)^(3/2))^(1/3)*(2*_a^4+3*_a^2+1)), _a = _Z)+2*c__1)*x

> 

dsolve(ode,y(x),[homogeneous])

Classification methods on request

Methods to be used are: [homogeneous]

Successful isolation of dy/dx: 3 solutions were found. Trying to solve each resulting ODE.

----------------------------

* Tackling ODE using method: homogeneous

--- Trying classification methods ---

trying homogeneous types:

> 

sol:=PDEtools:-Solve(ode,diff(y(x),x));

diff(y(x), x) = (1/2)*2^(1/3)*(-y(x)^2*2^(1/3)+((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(2/3))/(y(x)*((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(1/3)), diff(y(x), x) = -(1/4)*2^(1/3)*(I*3^(1/2)*y(x)^2*2^(1/3)+I*3^(1/2)*((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(2/3)-y(x)^2*2^(1/3)+((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(2/3))/(y(x)*((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(1/3)), diff(y(x), x) = (1/4)*2^(1/3)*(I*3^(1/2)*y(x)^2*2^(1/3)+I*3^(1/2)*((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(2/3)+y(x)^2*2^(1/3)-((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(2/3))/(y(x)*((-x+(2*y(x)^2+x^2)^(1/2))*y(x)^2)^(1/3))

> 

map(X->DEtools:-odeadvisor(X),[sol])

[[[_homogeneous, `class A`], _dAlembert], [[_homogeneous, `class A`]], [[_homogeneous, `class A`]]]

> 

map(X->dsolve(X,y(x),[homogeneous]),[sol])

Classification methods on request

Methods to be used are: [homogeneous]

----------------------------

* Tackling ODE using method: homogeneous

--- Trying classification methods ---

trying homogeneous types:

Classification methods on request

Methods to be used are: [homogeneous]

----------------------------

* Tackling ODE using method: homogeneous

--- Trying classification methods ---

trying homogeneous types:

Classification methods on request

Methods to be used are: [homogeneous]

----------------------------

* Tackling ODE using method: homogeneous

--- Trying classification methods ---

trying homogeneous types:

[]

 

 

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