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ODEs Having Linear Symmetries

Description

• 

The general forms of ODEs having one of the following linear symmetries

[xi=a+b*x, eta=0], [xi=a+b*y, eta=0], [xi=0, eta=c+d*x], [xi=0, eta=c+d*y]:

• 

where the infinitesimal symmetry generator is given by:

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

G:=f→ξxf+ηyf

(1)
• 

 

• 

are given by:

ode[1] := DEtools[equinv]([xi=a+b*x, eta=0], y(x), 2);

ode1:=ⅆ2ⅆx2yx=_F1yx,ⅆⅆxyxbx+abx+a2

(2)

ode[2] := DEtools[equinv]([xi=a+b*y, eta=0], y(x), 2);

ode2:=ⅆ2ⅆx2yx=_F1yx,ⅆⅆxyxbxbyxaⅆⅆxyxbyx+aⅆⅆxyx3byx+a3

(3)

ode[3] := DEtools[equinv]([xi=0, eta=c+d*x], y(x), 2);

ode3:=ⅆ2ⅆx2yx=_F1x,ⅆⅆxyxdx+ⅆⅆxyxcdyxdx+c

(4)

ode[4] := DEtools[equinv]([xi=0, eta=c+d*y], y(x), 2);

ode4:=ⅆ2ⅆx2yx=_F1x,ⅆⅆxyxdyx+cdyx+_F1x,ⅆⅆxyxdyx+cc

(5)
  

Although the symmetries of these families of ODEs can be determined in a direct manner (using symgen), the simplicity of their pattern motivated us to have separate routines for recognizing them.

Examples

withDEtools,equinv,odeadvisor,symgen:

odeadvisorode1

_2nd_order,_with_linear_symmetries

(6)

odeadvisorode2

_2nd_order,_with_linear_symmetries

(7)

odeadvisorode3

_2nd_order,_with_linear_symmetries

(8)

odeadvisorode4

_2nd_order,_with_linear_symmetries

(9)

As an example that can be solved by the related routine, consider

ode5:=equinv0,y,x,0,yx,2

ode5:=ⅆ2ⅆx2yx=_F1ⅆⅆxyxxyxyxx2

(10)

dsolveode5

yx=ⅇ∫lnxRootOf∫_Z1_a_a2+_F1_aⅆ_a_b+_C1ⅆ_b+_C2

(11)

See Also

DEtools, odeadvisor, dsolve,Lie, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.


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