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DEtools

  

buildsym

  

build the symmetry generator given a solution of an ODE

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

buildsym(sol, y(x), _C1, _C2)

Parameters

sol

-

expression representing the solution of an ODE

y(x)

-

(optional) dependent variable

_C1, _C2

-

(optional) arbitrary constants of the ODE solution

Description

• 

The buildsym command takes as arguments the solution of an ODE and the dependent variable, and it looks for a pair of infinitesimals for the ODE that generated the problem. At present, buildsym is able to look for symmetries of solutions of first and second order ODEs only.

• 

As a rule, if we represent the infinitesimal generator as

f -> xi(x,y) * diff(f,x) + eta(x,y) * diff(f,y);

f→ξx,yxf+ηx,yyf

(1)
  

where x is the independent variable and y is the dependent one, buildsym (when successful) returns the list ξx,y,ηx,y, or two of these lists if the solution given as argument contains two arbitrary constants _C1 and _C2.

• 

In order to study the relationship between symmetry patterns and ODE patterns, the buildsym command and equinv may be useful in connection with the odeadvisor command and the HINT=e1,e2 option of dsolve.

• 

This function is part of the DEtools package, and so it can be used in the form buildsym(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[buildsym](..).

Examples

withDEtools,buildsym,equinv,symtest

buildsym,equinv,symtest

(2)

The complete symmetry cycle

1) A pair of infinitesimals

sym_ξ=0,_η=ⅇylnx

sym:=_ξ=0,_η=ⅇylnx

(3)

The most general first order ODE invariant under the above is as follows:

ODEequinvsym,yx

ODE:=ⅆⅆxyx=1lnxx+ⅇyx_F1x

(4)

This ODE can be solved using the following:

ansdsolveODE,Lie

ans:=yx=ln∫_F1xlnxⅆx+_C1lnx

(5)

The infinitesimals can be reobtained from the solution above:

buildsymans,yx

_ξ=0,_η=ⅇylnx

(6)

2) Given the solution of a second order ODE, for instance, we get the following:

ans2yx=_C1cosx+x_C2

ans2:=yx=_C1cosx+x_C2

(7)

The infinitesimals of an underlying Lie symmetry group

sym2buildsymans2

sym2:=_ξ=0,_η=x,_ξ=0,_η=cosx

(8)

The most general second order ODE possessing these symmetries

ODE2equinvsym2,yx,2

ODE2:=ⅆ2ⅆx2yx=xcosxⅆⅆxyxsinxx+cosxcosxyxsinxx+cosx+_F1x

(9)

To test symmetries, one can use the symtest command; sym2[1] is a symmetry of ODE2:

symtestsym21,ODE2

0

(10)

sym2[2] is a symmetry of ODE2:

symtestsym22,ODE2

0

(11)

See Also

DEtools

dsolve

dsolve,Lie

equinv

odeadvisor

odetest

PDEtools

symgen

symtest

 


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