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DEtools

 ODEInvariants
 computes relative invariants for linear and nonlinear ODEs of order 3 and higher

 Calling Sequence ODEInvariants(ODE, y(x))

Parameters

 ODE - ordinary differential equation satisfied by y(x) y(x) - (optional) dependent variable; required when the ODE contains more than one function being differentiated

Description

 • 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 form$x\to F\left(x\right),y\left(x\right)\to P\left(x\right)y\left(x\right)$. The weight of each of these relative invariants is given by the power of the derivative of $F\left(x\right)$ entering as a factor in the transformed invariant, and given two relative invariants ${I}_{r}$ and ${I}_{s}$ respectively of weights $r$ and $s$, an absolute invariant can be constructed by taking $\frac{{I}_{r}^{s}}{{I}_{s}^{r}}$ (see references [1] and [2]).
 • 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 [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,

 ${y}^{\left(m\right)}+c[m-2]\left(x\right){y}^{\left(m-2\right)}+\mathrm{...}+c[1]\left(x\right)y\text{'}+c[0]\left(x\right)y=0$ (1)

 by transforming this equation using

$\left\{x\to F\left(x\right),y\to {\left(F\text{'}\right)}^{\frac{m-1}{2}}u\left(t\right)\right\}$

 we obtain an equation of the same form as (1). Performing now a sequential reduction of the transformed $c[m-j]\left(x\right)$ coefficients, $j=3..m$, eliminating derivatives of $F\left(x\right)$, 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 $y\left(x\right)$ and its derivatives, an auxiliary linear ODE is constructed - say in $u\left(x\right)$ - where the coefficient of each derivative of $u\left(x\right)$ in this linear ODE is equal to the coefficient of the derivative of $y\left(x\right)$ of the same order in the given nonlinear ODE. Thus, because the ODE in $y\left(x\right)$ is nonlinear, this auxiliary linear ODE in $u\left(x\right)$ has coefficients involving $y\left(x\right)$ and its derivatives. Next the Wilczynski invariants are computed for this linear ODE in $u\left(x\right)$ and finally they are reduced with respect to the given nonlinear ODE in $y\left(x\right)$ (i.e., the mth derivative of $y\left(x\right)$ is isolated and replaced in the invariants).
 • Note that in the nonlinear case the invariants may dependent on the unknown $y\left(x\right)$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{ODEInvariants}\right)$
 $\left[{\mathrm{ODEInvariants}}\right]$ (1)
 > $\mathrm{PDEtools}:-\mathrm{declare}\left(\left(C,F,c,u,y\right)\left(x\right),\mathrm{prime}=x\right)$
 ${C}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{C}$
 ${F}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{F}$
 ${c}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{c}$
 ${u}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{u}$
 ${y}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{y}$
 ${\mathrm{derivatives with respect to}}{}{x}{}{\mathrm{of functions of one variable will now be displayed with \text{'}}}$ (2)

Consider the general form of a third order linear ODE

 > ${\mathrm{ode}}_{3}≔\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)=\mathrm{add}\left(c[j]\left(x\right)\left(\frac{{ⅆ}^{j}}{ⅆ{x}^{j}}y\left(x\right)\right),j=\left[2,1,0\right]\right)$
 ${{\mathrm{ode}}}_{{3}}{≔}{\mathrm{y\text{'}\text{'}\text{'}}}{=}{y}{}{{c}}_{{0}}{+}{\mathrm{y\text{'}}}{}{{c}}_{{1}}{+}{\mathrm{y\text{'}\text{'}}}{}{{c}}_{{2}}$ (3)

For ODEs of third order ODEInvariants returns one invariant

 > $\mathrm{ODEInvariants}\left({\mathrm{ode}}_{3}\right)$
 $\left[{2}{}{{c}}_{{0}}{+}\frac{{2}}{{3}}{}{{c}}_{{1}}{}{{c}}_{{2}}{+}\frac{{4}}{{27}}{}{{c}}_{{2}}^{{3}}{+}\frac{{1}}{{3}}{}{{c}}_{{2}}^{{\mathrm{\text{'}\text{'}}}}{-}{{c}}_{{1}}^{{\mathrm{\text{'}}}}{-}\frac{{2}}{{3}}{}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{}{{c}}_{{2}}\right]$ (4)

Let's check that the returned invariants are relative invariants in the case of a  fourth order linear ODE

 > ${\mathrm{ode}}_{4}≔\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}y\left(x\right)=\mathrm{add}\left(c[j]\left(x\right)\left(\frac{{ⅆ}^{j}}{ⅆ{x}^{j}}y\left(x\right)\right),j=\left[3,2,1,0\right]\right)$
 ${{\mathrm{ode}}}_{{4}}{≔}{\mathrm{y\text{'}\text{'}\text{'}\text{'}}}{=}{y}{}{{c}}_{{0}}{+}{\mathrm{y\text{'}}}{}{{c}}_{{1}}{+}{\mathrm{y\text{'}\text{'}}}{}{{c}}_{{2}}{+}{\mathrm{y\text{'}\text{'}\text{'}}}{}{{c}}_{{3}}$ (5)
 > $\mathrm{ii}≔\mathrm{ODEInvariants}\left({\mathrm{ode}}_{4}\right)$
 ${\mathrm{ii}}{≔}\left[{{c}}_{{1}}{+}\frac{{1}}{{2}}{}{{c}}_{{2}}{}{{c}}_{{3}}{+}\frac{{1}}{{8}}{}{{c}}_{{3}}^{{3}}{+}\frac{{1}}{{2}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{-}\frac{{3}}{{4}}{}{{c}}_{{3}}{}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{,}\frac{{1}}{{4}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}\text{'}}}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}\text{'}}}}{-}\frac{{3}}{{4}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{}{{c}}_{{3}}{-}\frac{{33}}{{40}}{}{\left({{c}}_{{3}}^{{\mathrm{\text{'}}}}\right)}^{{2}}{+}\frac{{1}}{{320}}{}\left({312}{}{{c}}_{{3}}^{{2}}{+}{432}{}{{c}}_{{2}}\right){}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{-}\frac{{39}}{{320}}{}{{c}}_{{3}}^{{4}}{-}\frac{{13}}{{20}}{}{{c}}_{{3}}^{{2}}{}{{c}}_{{2}}{-}\frac{{5}}{{4}}{}{{c}}_{{1}}{}{{c}}_{{3}}{-}\frac{{9}}{{20}}{}{{c}}_{{2}}^{{2}}{+}\frac{{5}}{{4}}{}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{}{{c}}_{{3}}{-}{5}{}{{c}}_{{0}}{+}\frac{{5}}{{2}}{}{{c}}_{{1}}^{{\mathrm{\text{'}}}}\right]$ (6)

By definition, these expressions are relative invariants if when we transform in them the coefficients c[j](x) using

 > $\mathrm{tr}≔\left\{x=F\left(t\right),y\left(x\right)={\left(\frac{ⅆ}{ⅆt}F\left(t\right)\right)}^{\frac{3}{2}}u\left(t\right)\right\}$
 ${\mathrm{tr}}{≔}\left\{{x}{=}{F}{}\left({t}\right){,}{y}{=}{{F}}_{{t}}^{\left({3}{/}{2}\right)}{}{u}{}\left({t}\right)\right\}$ (7)

the resulting expressions are of the form ${\left(F\mathrm{\text{'} }\right)}^{k}\mathrm{\Phi }\left({c}_{j}\left(F\right)\right)$, and if next, by replacing F by the identity, we reobtain the departing expressions $\mathrm{ii}$

So we proceed first transforming these coefficients entering $\mathrm{ii}$ and for that purpose transform ode[4]

 > $\mathrm{PDEtools}:-\mathrm{dchange}\left(\mathrm{tr},{\mathrm{ode}}_{4},\left[t,u\left(t\right)\right],\mathrm{known}=\mathrm{all},\mathrm{simplify}\right)$
 ${-}\frac{{1}}{{16}}{}\frac{{135}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}^{{4}}{-}{300}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}^{{2}}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}{+}{120}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}{}{{F}}_{{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{+}{60}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}{,}{t}}^{{2}}{}{{F}}_{{t}}^{{2}}{-}{24}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{-}{240}{}{{F}}_{{t}{,}{t}}^{{3}}{}{{u}}_{{t}}{}{{F}}_{{t}}{+}{120}{}{{F}}_{{t}{,}{t}}^{{2}}{}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{+}{320}{}{{F}}_{{t}{,}{t}}{}{{u}}_{{t}}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{-}{80}{}{{u}}_{{t}}{}{{F}}_{{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{-}{80}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{-}{16}{}{{u}}_{{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{4}}}{{{F}}_{{t}}^{\left({13}{/}{2}\right)}}{=}\frac{{1}}{{8}}{}\frac{{8}{}{c}{[}{0}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}}^{{6}}{+}{12}{}{c}{[}{1}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{4}}{+}{8}{}{c}{[}{1}{]}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}}{}{{F}}_{{t}}^{{5}}{-}{6}{}{c}{[}{2}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}^{{2}}{}{{F}}_{{t}}^{{2}}{+}{12}{}{c}{[}{2}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{+}{16}{}{c}{[}{2}{]}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}{,}{t}}{}{{u}}_{{t}}{}{{F}}_{{t}}^{{3}}{+}{8}{}{c}{[}{2}{]}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{4}}{+}{15}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}^{{3}}{-}{30}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}{+}{12}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{-}{30}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}{,}{t}}^{{2}}{}{{u}}_{{t}}{}{{F}}_{{t}}{+}{12}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}{,}{t}}{}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{+}{28}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{+}{8}{}{c}{[}{3}{]}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}}{{{F}}_{{t}}^{\left({9}{/}{2}\right)}}$ (8)

To get the transformed coefficients $C[j]\left(x\right)$, first isolate u''''

 > $\mathrm{subs}\left(t=x,\mathrm{isolate}\left(,\frac{{ⅆ}^{4}}{ⅆ{t}^{4}}u\left(t\right)\right)\right)$
 ${\mathrm{u\text{'}\text{'}\text{'}\text{'}}}{=}{-}\frac{{1}}{{16}}{}\frac{{-}{2}{}{{\mathrm{F\text{'}}}}^{{2}}{}\left({8}{}{c}{[}{0}{]}{}\left({F}\right){}{u}{}{{\mathrm{F\text{'}}}}^{{6}}{+}{12}{}{c}{[}{1}{]}{}\left({F}\right){}{u}{}{\mathrm{F\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{4}}{+}{8}{}{c}{[}{1}{]}{}\left({F}\right){}{\mathrm{u\text{'}}}{}{{\mathrm{F\text{'}}}}^{{5}}{-}{6}{}{c}{[}{2}{]}{}\left({F}\right){}{u}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{{\mathrm{F\text{'}}}}^{{2}}{+}{12}{}{c}{[}{2}{]}{}\left({F}\right){}{u}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{16}{}{c}{[}{2}{]}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{u\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{8}{}{c}{[}{2}{]}{}\left({F}\right){}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{4}}{+}{15}{}{c}{[}{3}{]}{}\left({F}\right){}{u}{}{{\mathrm{F\text{'}\text{'}}}}^{{3}}{-}{30}{}{c}{[}{3}{]}{}\left({F}\right){}{u}{}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{\mathrm{F\text{'}}}{+}{12}{}{c}{[}{3}{]}{}\left({F}\right){}{u}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}{-}{30}{}{c}{[}{3}{]}{}\left({F}\right){}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}}}{+}{12}{}{c}{[}{3}{]}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}{+}{28}{}{c}{[}{3}{]}{}\left({F}\right){}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}{+}{8}{}{c}{[}{3}{]}{}\left({F}\right){}{\mathrm{u\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}\right){-}{135}{}{u}{}{{\mathrm{F\text{'}\text{'}}}}^{{4}}{+}{300}{}{u}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{\mathrm{F\text{'}}}{-}{120}{}{u}{}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}{-}{60}{}{u}{}{{\mathrm{F\text{'}\text{'}\text{'}}}}^{{2}}{}{{\mathrm{F\text{'}}}}^{{2}}{+}{24}{}{u}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{240}{}{{\mathrm{F\text{'}\text{'}}}}^{{3}}{}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}}}{-}{120}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}{-}{320}{}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}{+}{80}{}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{80}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}}{{{\mathrm{F\text{'}}}}^{{4}}}$ (9)

Compute now the coefficients $C[j]\left(x\right)$ of derivatives of $u\left(x\right)$ in the transformed equation

 > $\mathrm{zip}\left(\mathrm{=},\left[C[3]\left(x\right),C[2]\left(x\right),C[1]\left(x\right),C[0]\left(x\right)\right],\left[\mathrm{PDEtools}:-\mathrm{dcoeffs}\left(\mathrm{rhs}\left(\right),u\left(x\right)\right)\right]\right)$
 $\left[{{C}}_{{3}}{=}{\mathrm{F\text{'}}}{}{c}{[}{3}{]}{}\left({F}\right){,}{{C}}_{{2}}{=}{{\mathrm{F\text{'}}}}^{{2}}{}{c}{[}{2}{]}{}\left({F}\right){+}\frac{{3}}{{2}}{}{\mathrm{F\text{'}\text{'}}}{}{c}{[}{3}{]}{}\left({F}\right){-}\frac{{5}{}{\mathrm{F\text{'}\text{'}\text{'}}}}{{\mathrm{F\text{'}}}}{+}\frac{{15}}{{2}}{}\frac{{{\mathrm{F\text{'}\text{'}}}}^{{2}}}{{{\mathrm{F\text{'}}}}^{{2}}}{,}{{C}}_{{1}}{=}{{\mathrm{F\text{'}}}}^{{3}}{}{c}{[}{1}{]}{}\left({F}\right){+}{2}{}{\mathrm{F\text{'}}}{}{\mathrm{F\text{'}\text{'}}}{}{c}{[}{2}{]}{}\left({F}\right){+}\frac{{7}}{{2}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{c}{[}{3}{]}{}\left({F}\right){-}\frac{{15}}{{4}}{}\frac{{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{c}{[}{3}{]}{}\left({F}\right)}{{\mathrm{F\text{'}}}}{-}\frac{{5}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}}{{\mathrm{F\text{'}}}}{+}\frac{{20}{}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}}}}{{{\mathrm{F\text{'}}}}^{{2}}}{-}\frac{{15}{}{{\mathrm{F\text{'}\text{'}}}}^{{3}}}{{{\mathrm{F\text{'}}}}^{{3}}}{,}{{C}}_{{0}}{=}{{\mathrm{F\text{'}}}}^{{4}}{}{c}{[}{0}{]}{}\left({F}\right){+}\frac{{3}}{{2}}{}{{\mathrm{F\text{'}}}}^{{2}}{}{c}{[}{1}{]}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}{+}\frac{{3}}{{2}}{}{\mathrm{F\text{'}}}{}{c}{[}{2}{]}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}\text{'}}}{-}\frac{{3}}{{4}}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{c}{[}{2}{]}{}\left({F}\right){+}\frac{{3}}{{2}}{}{c}{[}{3}{]}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{-}\frac{{15}}{{4}}{}\frac{{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{c}{[}{3}{]}{}\left({F}\right)}{{\mathrm{F\text{'}}}}{+}\frac{{15}}{{8}}{}\frac{{{\mathrm{F\text{'}\text{'}}}}^{{3}}{}{c}{[}{3}{]}{}\left({F}\right)}{{{\mathrm{F\text{'}}}}^{{2}}}{-}\frac{{3}}{{2}}{}\frac{{\mathrm{F\text{'}\text{'}\text{'}\text{'}\text{'}}}}{{\mathrm{F\text{'}}}}{+}\frac{{15}}{{2}}{}\frac{{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}}{{{\mathrm{F\text{'}}}}^{{2}}}{+}\frac{{15}}{{4}}{}\frac{{{\mathrm{F\text{'}\text{'}\text{'}}}}^{{2}}}{{{\mathrm{F\text{'}}}}^{{2}}}{-}\frac{{75}}{{4}}{}\frac{{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{\mathrm{F\text{'}\text{'}\text{'}}}}{{{\mathrm{F\text{'}}}}^{{3}}}{+}\frac{{135}}{{16}}{}\frac{{{\mathrm{F\text{'}\text{'}}}}^{{4}}}{{{\mathrm{F\text{'}}}}^{{4}}}\right]$ (10)

Compute now the invariants $\mathrm{ii}$ using these coefficients $C[j]\left(x\right)$ expressed in terms of the $c[j]\left(x\right)$ using the formula above

 > $\mathrm{subs}\left(c=C,\mathrm{ii}\right)$
 $\left[{{C}}_{{1}}{+}\frac{{1}}{{2}}{}{{C}}_{{2}}{}{{C}}_{{3}}{+}\frac{{1}}{{8}}{}{{C}}_{{3}}^{{3}}{+}\frac{{1}}{{2}}{}{{C}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{-}{{C}}_{{2}}^{{\mathrm{\text{'}}}}{-}\frac{{3}}{{4}}{}{{C}}_{{3}}{}{{C}}_{{3}}^{{\mathrm{\text{'}}}}{,}\frac{{1}}{{4}}{}{{C}}_{{3}}^{{\mathrm{\text{'}\text{'}\text{'}}}}{-}{{C}}_{{2}}^{{\mathrm{\text{'}\text{'}}}}{-}\frac{{3}}{{4}}{}{{C}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{}{{C}}_{{3}}{-}\frac{{33}}{{40}}{}{\left({{C}}_{{3}}^{{\mathrm{\text{'}}}}\right)}^{{2}}{+}\frac{{1}}{{320}}{}\left({312}{}{{C}}_{{3}}^{{2}}{+}{432}{}{{C}}_{{2}}\right){}{{C}}_{{3}}^{{\mathrm{\text{'}}}}{-}\frac{{39}}{{320}}{}{{C}}_{{3}}^{{4}}{-}\frac{{13}}{{20}}{}{{C}}_{{3}}^{{2}}{}{{C}}_{{2}}{-}\frac{{5}}{{4}}{}{{C}}_{{1}}{}{{C}}_{{3}}{-}\frac{{9}}{{20}}{}{{C}}_{{2}}^{{2}}{+}\frac{{5}}{{4}}{}{{C}}_{{2}}^{{\mathrm{\text{'}}}}{}{{C}}_{{3}}{-}{5}{}{{C}}_{{0}}{+}\frac{{5}}{{2}}{}{{C}}_{{1}}^{{\mathrm{\text{'}}}}\right]$ (11)
 > $\mathrm{factor}\left(\mathrm{eval}\left(,\right)\right)$
 $\left[\frac{{1}}{{8}}{}{{\mathrm{F\text{'}}}}^{{3}}{}\left({{c}{[}{3}{]}{}\left({F}\right)}^{{3}}{-}{6}{}{\mathrm{D}}{}\left({c}{[}{3}{]}\right){}\left({F}\right){}{c}{[}{3}{]}{}\left({F}\right){+}{4}{}{c}{[}{2}{]}{}\left({F}\right){}{c}{[}{3}{]}{}\left({F}\right){+}{4}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({c}{[}{3}{]}\right){}\left({F}\right){-}{8}{}{\mathrm{D}}{}\left({c}{[}{2}{]}\right){}\left({F}\right){+}{8}{}{c}{[}{1}{]}{}\left({F}\right)\right){,}\frac{{1}}{{320}}{}{{\mathrm{F\text{'}}}}^{{4}}{}\left({-}{39}{}{{c}{[}{3}{]}{}\left({F}\right)}^{{4}}{+}{312}{}{\mathrm{D}}{}\left({c}{[}{3}{]}\right){}\left({F}\right){}{{c}{[}{3}{]}{}\left({F}\right)}^{{2}}{-}{208}{}{c}{[}{2}{]}{}\left({F}\right){}{{c}{[}{3}{]}{}\left({F}\right)}^{{2}}{-}{240}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({c}{[}{3}{]}\right){}\left({F}\right){}{c}{[}{3}{]}{}\left({F}\right){+}{400}{}{\mathrm{D}}{}\left({c}{[}{2}{]}\right){}\left({F}\right){}{c}{[}{3}{]}{}\left({F}\right){-}{264}{}{{\mathrm{D}}{}\left({c}{[}{3}{]}\right){}\left({F}\right)}^{{2}}{+}{432}{}{\mathrm{D}}{}\left({c}{[}{3}{]}\right){}\left({F}\right){}{c}{[}{2}{]}{}\left({F}\right){-}{400}{}{c}{[}{1}{]}{}\left({F}\right){}{c}{[}{3}{]}{}\left({F}\right){-}{144}{}{{c}{[}{2}{]}{}\left({F}\right)}^{{2}}{+}{800}{}{\mathrm{D}}{}\left({c}{[}{1}{]}\right){}\left({F}\right){+}{80}{}{{\mathrm{D}}}^{\left({3}\right)}{}\left({c}{[}{3}{]}\right){}\left({F}\right){-}{320}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({c}{[}{2}{]}\right){}\left({F}\right){-}{1600}{}{c}{[}{0}{]}{}\left({F}\right)\right)\right]$ (12)

It is visible that each expression is now of the form ${\left(F\mathrm{\text{'} }\right)}^{k}\mathrm{\Phi }\left({c}_{j}\left(F\right)\right)$, and according to the description, the first relative invariant has weight 3 (in the factor ${\left(F\text{'}\right)}^{k},k=3$) and the second one has weight 4. Let's verify that at $F=\mathrm{identity}$ we reobtain the departing expressions ii, proving in that way that the expressions ii are relative invariants

 > $\mathrm{convert}\left(\genfrac{}{}{0}{}{\phantom{F=\left(x→x\right)}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{}}{F=\left(x→x\right)},\mathrm{diff}\right)$
 $\left[{{c}}_{{1}}{+}\frac{{1}}{{2}}{}{{c}}_{{2}}{}{{c}}_{{3}}{+}\frac{{1}}{{8}}{}{{c}}_{{3}}^{{3}}{+}\frac{{1}}{{2}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{-}\frac{{3}}{{4}}{}{{c}}_{{3}}{}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{,}{-}\frac{{39}}{{320}}{}{{c}}_{{3}}^{{4}}{+}\frac{{39}}{{40}}{}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{}{{c}}_{{3}}^{{2}}{-}\frac{{13}}{{20}}{}{{c}}_{{3}}^{{2}}{}{{c}}_{{2}}{-}\frac{{3}}{{4}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{}{{c}}_{{3}}{-}\frac{{5}}{{4}}{}{{c}}_{{1}}{}{{c}}_{{3}}{+}\frac{{5}}{{4}}{}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{}{{c}}_{{3}}{-}\frac{{33}}{{40}}{}{\left({{c}}_{{3}}^{{\mathrm{\text{'}}}}\right)}^{{2}}{+}\frac{{27}}{{20}}{}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{}{{c}}_{{2}}{-}\frac{{9}}{{20}}{}{{c}}_{{2}}^{{2}}{+}\frac{{1}}{{4}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}\text{'}}}}{+}\frac{{5}}{{2}}{}{{c}}_{{1}}^{{\mathrm{\text{'}}}}{-}{5}{}{{c}}_{{0}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}\text{'}}}}\right]$ (13)
 > $\mathrm{normal}\left(-\mathrm{ii}\right)$
 $\left[{0}{,}{0}\right]$ (14)

Let's now transform the linear equation ode[4] into a nonlinear one by means of a point transformation

 > $\mathrm{PDEtools}:-\mathrm{dchange}\left(y\left(x\right)=\frac{1}{u\left(x\right)},{\mathrm{ode}}_{4},\left[u\left(x\right)\right]\right)$
 $\frac{{24}{}{{\mathrm{u\text{'}}}}^{{4}}}{{{u}}^{{5}}}{-}\frac{{36}{}{{\mathrm{u\text{'}}}}^{{2}}{}{\mathrm{u\text{'}\text{'}}}}{{{u}}^{{4}}}{+}\frac{{6}{}{{\mathrm{u\text{'}\text{'}}}}^{{2}}}{{{u}}^{{3}}}{+}\frac{{8}{}{\mathrm{u\text{'}}}{}{\mathrm{u\text{'}\text{'}\text{'}}}}{{{u}}^{{3}}}{-}\frac{{\mathrm{u\text{'}\text{'}\text{'}\text{'}}}}{{{u}}^{{2}}}{=}{{c}}_{{3}}{}\left({-}\frac{{6}{}{{\mathrm{u\text{'}}}}^{{3}}}{{{u}}^{{4}}}{+}\frac{{6}{}{\mathrm{u\text{'}}}{}{\mathrm{u\text{'}\text{'}}}}{{{u}}^{{3}}}{-}\frac{{\mathrm{u\text{'}\text{'}\text{'}}}}{{{u}}^{{2}}}\right){+}{{c}}_{{2}}{}\left(\frac{{2}{}{{\mathrm{u\text{'}}}}^{{2}}}{{{u}}^{{3}}}{-}\frac{{\mathrm{u\text{'}\text{'}}}}{{{u}}^{{2}}}\right){-}\frac{{{c}}_{{1}}{}{\mathrm{u\text{'}}}}{{{u}}^{{2}}}{+}\frac{{{c}}_{{0}}}{{u}}$ (15)
 > $\mathrm{nonlinearODE}≔\mathrm{isolate}\left(,\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}u\left(x\right)\right)$
 ${\mathrm{nonlinearODE}}{≔}{\mathrm{u\text{'}\text{'}\text{'}\text{'}}}{=}{-}\left({{c}}_{{3}}{}\left({-}\frac{{6}{}{{\mathrm{u\text{'}}}}^{{3}}}{{{u}}^{{4}}}{+}\frac{{6}{}{\mathrm{u\text{'}}}{}{\mathrm{u\text{'}\text{'}}}}{{{u}}^{{3}}}{-}\frac{{\mathrm{u\text{'}\text{'}\text{'}}}}{{{u}}^{{2}}}\right){+}{{c}}_{{2}}{}\left(\frac{{2}{}{{\mathrm{u\text{'}}}}^{{2}}}{{{u}}^{{3}}}{-}\frac{{\mathrm{u\text{'}\text{'}}}}{{{u}}^{{2}}}\right){-}\frac{{{c}}_{{1}}{}{\mathrm{u\text{'}}}}{{{u}}^{{2}}}{+}\frac{{{c}}_{{0}}}{{u}}{-}\frac{{24}{}{{\mathrm{u\text{'}}}}^{{4}}}{{{u}}^{{5}}}{+}\frac{{36}{}{{\mathrm{u\text{'}}}}^{{2}}{}{\mathrm{u\text{'}\text{'}}}}{{{u}}^{{4}}}{-}\frac{{6}{}{{\mathrm{u\text{'}\text{'}}}}^{{2}}}{{{u}}^{{3}}}{-}\frac{{8}{}{\mathrm{u\text{'}}}{}{\mathrm{u\text{'}\text{'}\text{'}}}}{{{u}}^{{3}}}\right){}{{u}}^{{2}}$ (16)
 > $\mathrm{ODEInvariants}\left(\mathrm{nonlinearODE}\right)$
 $\left[{{c}}_{{1}}{+}\frac{{1}}{{2}}{}{{c}}_{{2}}{}{{c}}_{{3}}{+}\frac{{1}}{{8}}{}{{c}}_{{3}}^{{3}}{+}\frac{{1}}{{2}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{-}\frac{{3}}{{4}}{}{{c}}_{{3}}{}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{,}\frac{{1}}{{4}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}\text{'}}}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}\text{'}}}}{-}\frac{{3}}{{4}}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{}{{c}}_{{3}}{-}\frac{{33}}{{40}}{}{\left({{c}}_{{3}}^{{\mathrm{\text{'}}}}\right)}^{{2}}{+}\frac{{1}}{{320}}{}\left({312}{}{{c}}_{{3}}^{{2}}{+}{432}{}{{c}}_{{2}}\right){}{{c}}_{{3}}^{{\mathrm{\text{'}}}}{-}\frac{{39}}{{320}}{}{{c}}_{{3}}^{{4}}{-}\frac{{13}}{{20}}{}{{c}}_{{3}}^{{2}}{}{{c}}_{{2}}{-}\frac{{5}}{{4}}{}{{c}}_{{1}}{}{{c}}_{{3}}{-}\frac{{9}}{{20}}{}{{c}}_{{2}}^{{2}}{+}\frac{{5}}{{4}}{}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{}{{c}}_{{3}}{-}{5}{}{{c}}_{{0}}{+}\frac{{5}}{{2}}{}{{c}}_{{1}}^{{\mathrm{\text{'}}}}\right]$ (17)

The expressions above depend only on $x$, not on $u\left(x\right)$ or its derivatives, because this nonlinear ODE above is related - by construction - to a linear ODE (ode[4]) through a point transformation ($y\to \frac{1}{y}$ 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:

 > $\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}u\left(x\right)={u\left(x\right)}^{3}\left(\frac{ⅆ}{ⅆx}u\left(x\right)\right)+{\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}u\left(x\right)\right)}^{2}+x$
 ${\mathrm{u\text{'}\text{'}\text{'}\text{'}}}{=}{{u}}^{{3}}{}{\mathrm{u\text{'}}}{+}{{\mathrm{u\text{'}\text{'}}}}^{{2}}{+}{x}$ (18)
 > $\mathrm{ODEInvariants}\left(\right)$
 $\left[{{u}}^{{3}}{-}{2}{}{\mathrm{u\text{'}\text{'}\text{'}}}{,}{-}\frac{{15}}{{2}}{}{\mathrm{u\text{'}}}{}{{u}}^{{2}}{-}\frac{{19}}{{5}}{}{{\mathrm{u\text{'}\text{'}}}}^{{2}}{-}{2}{}{{u}}^{{3}}{}{\mathrm{u\text{'}}}{-}{2}{}{x}\right]$ (19)

References

 [1] Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.
 [2] Chalkley, R., Basic Global Relative Invariants for Homogeneous Linear Differential Equations, Amer Mathematical Society (2002).
 [3] Wilczynski, E.J., Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905.