ODEs Having Linear Symmetries
Description
Examples
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→ξ∂∂xf+η∂∂yf
are given by:
ode[1] := DEtools[equinv]([xi=a+b*x, eta=0], y(x), 2);
ode1≔ⅆ2ⅆx2yx=f__1yx,ⅆⅆxyxbx+abx+a2
ode[2] := DEtools[equinv]([xi=a+b*y, eta=0], y(x), 2);
ode2≔ⅆ2ⅆx2yx=f__1yx,−ⅆⅆxyxbx−byx−aⅆⅆxyxbyx+aⅆⅆxyx3byx+a3
ode[3] := DEtools[equinv]([xi=0, eta=c+d*x], y(x), 2);
ode3≔ⅆ2ⅆx2yx=f__1x,ⅆⅆxyxdx+ⅆⅆxyxc−dyxxd+c
ode[4] := DEtools[equinv]([xi=0, eta=c+d*y], y(x), 2);
ode4≔ⅆ2ⅆx2yx=f__1x,ⅆⅆxyxdyx+cdyx+f__1x,ⅆⅆxyxdyx+cc
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.
withDEtools,equinv,odeadvisor,symgen:
odeadvisorode1
_2nd_order,_with_linear_symmetries
odeadvisorode2
odeadvisorode3
odeadvisorode4
As an example that can be solved by the related routine, consider
ode5≔equinv0,y,x,0,yx,2
ode5≔ⅆ2ⅆx2yx=f__1ⅆⅆxyxxyxyxx2
dsolveode5
yx=ⅇ∫` `lnxRootOf−∫` `_Z1−_a+_a2−f__1_aⅆ_a−_b+c__1ⅆ_b+c__2
See Also
DEtools
odeadvisor
dsolve,Lie
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
odeadvisor,types
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