computes relative invariants for linear and nonlinear ODEs of order 3 and higher
ordinary differential equation satisfied by y(x)
(optional) dependent variable; required when the ODE contains more than one function being differentiated
Given a linear or nonlinear ODE of order m=3 or higher, ODEInvariants returns a list of m−2 relative invariants under transformations of the formx→Fx,yx→Px⁢yx. The weight of each of these relative invariants is given by the power of the derivative of F⁡x entering as a factor in the transformed invariant, and given two relative invariants Ir and Is respectively of weights r and s, an absolute invariant can be constructed by taking IrsIsr (see references  and ).
The invariants in the returned list are ordered according to increasing weight, from weight = 3 to weight = m, the order of the equation. For example, for a fourth order ODE, the returned list contains two relative invariants, respectively of weights 3 and 4.
In the case of linear ODEs, these invariants coincide with the Wilczynski invariants (see reference ) although their computation is performed without rewriting the linear equation in Laguerre-Forsyth form. Instead, given a linear ODE of order 3 or higher, in normal form,
by transforming this equation using
we obtain an equation of the same form as (1). Performing now a sequential reduction of the transformed cm−j⁡x coefficients, j=3..m, eliminating derivatives of F⁡x, a sequence of expressions result that coincide with the Wilczynski relative invariants. The advantage of this process if that it does not require rewriting the linear equation in Laguerre-Forsyth form, which in turn would require solving a linear ODE of order m-1.
For nonlinear ODEs of order m=3 or higher, that are polynomial in the unknown y⁡x and its derivatives, an auxiliary linear ODE is constructed - say in u⁡x - where the coefficient of each derivative of u⁡x in this linear ODE is equal to the coefficient of the derivative of y⁡x of the same order in the given nonlinear ODE. Thus, because the ODE in y⁡x is nonlinear, this auxiliary linear ODE in u⁡x has coefficients involving y⁡x and its derivatives. Next the Wilczynski invariants are computed for this linear ODE in u⁡x and finally they are reduced with respect to the given nonlinear ODE in y⁡x (i.e., the mth derivative of y⁡x is isolated and replaced in the invariants).
Note that in the nonlinear case the invariants may dependent on the unknown y⁡x and its derivatives. However, if the nonlinear equation is linearizable through a point transformation these invariants will depend only on the independent variable x - see examples below.
C⁡x⁢will now be displayed as⁢C
F⁡x⁢will now be displayed as⁢F
c⁡x⁢will now be displayed as⁢c
u⁡x⁢will now be displayed as⁢u
y⁡x⁢will now be displayed as⁢y
derivatives with respect to⁢x⁢of functions of one variable will now be displayed with '
Consider the general form of a third order linear ODE
ode3 ≔ ⅆ3ⅆx3⁢y⁡x=add⁡cj⁡x⁢ⅆjⅆxj⁢y⁡x,j=2,1,0
For ODEs of third order ODEInvariants returns one invariant
Let's check that the returned invariants are relative invariants in the case of a fourth order linear ODE
ode4 ≔ ⅆ4ⅆx4⁢y⁡x=add⁡cj⁡x⁢ⅆjⅆxj⁢y⁡x,j=3,2,1,0
ii ≔ ODEInvariants⁡ode4
By definition, these expressions are relative invariants if when we transform in them the coefficients c[j](x) using
tr ≔ x=F⁡t,y⁡x=ⅆⅆt⁢F⁡t32⁢u⁡t
the resulting expressions are of the form F⁢' k⁢Φ⁡cj⁡F, and if next, by replacing F by the identity, we reobtain the departing expressions ii
So we proceed first transforming these coefficients entering ii and for that purpose transform ode
To get the transformed coefficients Cj⁡x, first isolate u''''
Compute now the coefficients Cj⁡x of derivatives of u⁡x in the transformed equation
Compute now the invariants ii using these coefficients Cj⁡x expressed in terms of the cj⁡x using the formula above
It is visible that each expression is now of the form F⁢' k⁢Φ⁡cj⁡F, and according to the description, the first relative invariant has weight 3 (in the factor F⁢' k,k=3) and the second one has weight 4. Let's verify that at F=identity we reobtain the departing expressions ii, proving in that way that the expressions ii are relative invariants
Let's now transform the linear equation ode into a nonlinear one by means of a point transformation
nonlinearODE ≔ isolate⁡,ⅆ4ⅆx4⁢u⁡x
The expressions above depend only on x, not on u⁡x or its derivatives, because this nonlinear ODE above is related - by construction - to a linear ODE (ode) through a point transformation (y→1y used above). Moreover: the invariants are the same as those in ii, of the related linear ode. When the nonlinear ODE cannot be related to a linear ODE through a point transformation, the invariants depend on the dependent variable and perhaps also its derivatives. For example:
 Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.
 Chalkley, R., Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Amer Mathematical Society (2002).
 Wilczynski, E.J., Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905.
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