DEtools[ODEInvariants] - computes relative invariants for linear and nonlinear ODEs of order 3 and higher
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Calling Sequence
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ODEInvariants(ODE, y(x))
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Parameters
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ODE
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ordinary differential equation satisfied by y(x)
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y(x)
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(optional) dependent variable; required when the ODE contains more than one function being differentiated
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Description
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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.
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In the case of linear ODEs, these invariants coincide with the Wilczynski invariants (see reference [3]) 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,
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 y^((m-2))+`...`+c[1](x) y `' `+c[0](x) y=0](/support/helpjp/helpview.aspx?si=8672/file01294/math85.png)
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(1)
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by transforming this equation using
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Note that in the nonlinear case the invariants may dependent on the unknown and its derivatives. However, if the nonlinear equation is linearizable through a point transformation these invariants will depend only on the independent variable - see examples below.
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Examples
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Consider the general form of a third order linear ODE
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For ODEs of third order ODEInvariants returns one invariant
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Let's check that the returned invariants are relative invariants in the case of a fourth order linear ODE
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![ii := [(1/2)*c[2]*c[3]+c[1]+(1/8)*c[3]^3+(1/2)*c[3]^`''`-(3/4)*c[3]^`'`*c[3]-c[2]^`'`, (1/4)*c[3]^`'''`-c[2]^`''`-(3/4)*c[3]^`''`*c[3]-(33/40)*(c[3]^`'`)^2+(1/320)*(432*c[2]+312*c[3]^2)*c[3]^`'`-(39/320)*c[3]^4-(5/4)*c[1]*c[3]-(13/20)*c[2]*c[3]^2-5*c[0]+(5/2)*c[1]^`'`+(5/4)*c[2]^`'`*c[3]-(9/20)*c[2]^2]](/support/helpjp/helpview.aspx?si=8672/file01294/math207.png)
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By definition, these expressions are relative invariants if when we transform in them the coefficients c[j](x) using
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the resulting expressions are of the form , and if next, by replacing F by the identity, we reobtain the departing expressions
So we proceed first transforming these coefficients entering and for that purpose transform ode[4]
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To get the transformed coefficients , first isolate u''''
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Compute now the coefficients of derivatives of in the transformed equation
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Compute now the invariants using these coefficients expressed in terms of the using the formula above
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![[(1/2)*C[2]*C[3]+C[1]+(1/8)*C[3]^3+(1/2)*C[3]^`''`-(3/4)*C[3]^`'`*C[3]-C[2]^`'`, (1/4)*C[3]^`'''`-C[2]^`''`-(3/4)*C[3]^`''`*C[3]-(33/40)*(C[3]^`'`)^2+(1/320)*(432*C[2]+312*C[3]^2)*C[3]^`'`-(39/320)*C[3]^4-(5/4)*C[1]*C[3]-(13/20)*C[2]*C[3]^2-5*C[0]+(5/2)*C[1]^`'`+(5/4)*C[2]^`'`*C[3]-(9/20)*C[2]^2]](/support/helpjp/helpview.aspx?si=8672/file01294/math276.png)
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*c[2](F)+8*c[1](F)+c[3](F)^3+4*((D@@2)(c[3]))(F)-6*c[3](F)*(D(c[3]))(F)-8*(D(c[2]))(F)), (1/320)*`F'`^4*(80*((D@@3)(c[3]))(F)-320*((D@@2)(c[2]))(F)+800*(D(c[1]))(F)-39*c[3](F)^4-1600*c[0](F)-144*c[2](F)^2-264*(D(c[3]))(F)^2-240*c[3](F)*((D@@2)(c[3]))(F)-400*c[3](F)*c[1](F)-208*c[3](F)^2*c[2](F)+400*c[3](F)*(D(c[2]))(F)+432*c[2](F)*(D(c[3]))(F)+312*c[3](F)^2*(D(c[3]))(F))]](/support/helpjp/helpview.aspx?si=8672/file01294/math283.png)
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It is visible that each expression is now of the form , and according to the description, the first relative invariant has weight 3 (in the factor ) and the second one has weight 4. Let's verify that at we reobtain the departing expressions ii, proving in that way that the expressions ii are relative invariants
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![[(1/2)*c[2]*c[3]+c[1]+(1/8)*c[3]^3+(1/2)*c[3]^`''`-(3/4)*c[3]^`'`*c[3]-c[2]^`'`, -(39/320)*c[3]^4+(39/40)*c[3]^`'`*c[3]^2+(1/4)*c[3]^`'''`-(33/40)*(c[3]^`'`)^2-5*c[0]-(5/4)*c[1]*c[3]-(13/20)*c[2]*c[3]^2+(27/20)*c[2]*c[3]^`'`-(9/20)*c[2]^2-(3/4)*c[3]^`''`*c[3]-c[2]^`''`+(5/4)*c[2]^`'`*c[3]+(5/2)*c[1]^`'`]](/support/helpjp/helpview.aspx?si=8672/file01294/math302.png)
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Let's now transform the linear equation ode[4] into a nonlinear one by means of a point transformation
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![24*`u'`^4/u^5-36*`u'`^2*`u''`/u^4+6*`u''`^2/u^3+8*`u'`*`u'''`/u^3-`u''''`/u^2 = c[3]*(-6*`u'`^3/u^4+6*`u'`*`u''`/u^3-`u'''`/u^2)+c[2]*(2*`u'`^2/u^3-`u''`/u^2)-c[1]*`u'`/u^2+c[0]/u](/support/helpjp/helpview.aspx?si=8672/file01294/math318.png)
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![nonlinearODE := `u''''` = -(c[3]*(-6*`u'`^3/u^4+6*`u'`*`u''`/u^3-`u'''`/u^2)+c[2]*(2*`u'`^2/u^3-`u''`/u^2)-c[1]*`u'`/u^2+c[0]/u-24*`u'`^4/u^5+36*`u'`^2*`u''`/u^4-6*`u''`^2/u^3-8*`u'`*`u'''`/u^3)*u^2](/support/helpjp/helpview.aspx?si=8672/file01294/math325.png)
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![[(1/2)*c[2]*c[3]+c[1]+(1/8)*c[3]^3+(1/2)*c[3]^`''`-(3/4)*c[3]^`'`*c[3]-c[2]^`'`, (1/4)*c[3]^`'''`-c[2]^`''`-(3/4)*c[3]^`''`*c[3]-(33/40)*(c[3]^`'`)^2+(1/320)*(432*c[2]+312*c[3]^2)*c[3]^`'`-(39/320)*c[3]^4-(5/4)*c[1]*c[3]-(13/20)*c[2]*c[3]^2-5*c[0]+(5/2)*c[1]^`'`+(5/4)*c[2]^`'`*c[3]-(9/20)*c[2]^2]](/support/helpjp/helpview.aspx?si=8672/file01294/math332.png)
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The expressions above depend only on , not on or its derivatives, because this nonlinear ODE above is related - by construction - to a linear ODE (ode[4]) through a point transformation ( used above). Moreover: the invariants are the same as those in ii, of the related linear ode[4]. 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:
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References
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[1] Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.
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[2] Chalkley, R., Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Amer Mathematical Society (2002).
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[3] Wilczynski, E.J., Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905.
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