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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  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,

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

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

For ODEs of third order ODEInvariants returns one invariant

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

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

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

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

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

the resulting expressions are of the form ${\left(F\text{'}\right)}^{k}\mathrm{\Phi }\left({c}_{j}\left(F\right)\right)$, 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

 > $\mathrm{PDEtools}:-\mathrm{dchange}\left(\mathrm{tr},{\mathrm{ode}}_{4},\left[t,u\left(t\right)\right],\mathrm{known}=\mathrm{all},\mathrm{simplify}\right)$
 $\frac{{16}{}{{u}}_{{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{4}}{+}{24}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{+}{80}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{+}{80}{}{{u}}_{{t}}{}{{F}}_{{t}{,}{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{-}{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}}{-}{120}{}{{F}}_{{t}{,}{t}}^{{2}}{}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{-}{320}{}{{F}}_{{t}{,}{t}}{}{{u}}_{{t}}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{2}}{+}{300}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}^{{2}}{}{{F}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}{+}{240}{}{{F}}_{{t}{,}{t}}^{{3}}{}{{u}}_{{t}}{}{{F}}_{{t}}{-}{135}{}{u}{}\left({t}\right){}{{F}}_{{t}{,}{t}}^{{4}}}{{16}{}{{F}}_{{t}}^{{13}}{{2}}}}{=}\frac{{3}{}{u}{}\left({t}\right){}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}{,}{t}{,}{t}{,}{t}}}{{2}{}{{F}}_{{t}}^{{5}}{{2}}}}{+}\frac{\left({12}{}{{c}}_{{2}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}}^{{3}}{}{u}{}\left({t}\right){-}{30}{}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}{,}{t}}{}{{F}}_{{t}}{}{u}{}\left({t}\right){+}{28}{}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}}{}{{F}}_{{t}}^{{2}}\right){}{{F}}_{{t}{,}{t}{,}{t}}{+}{8}{}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}{,}{t}{,}{t}}{}{{F}}_{{t}}^{{3}}{+}{15}{}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}{,}{t}}^{{3}}{}{u}{}\left({t}\right){+}\left({-}{6}{}{{c}}_{{2}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}}^{{2}}{}{u}{}\left({t}\right){-}{30}{}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}}{}{{F}}_{{t}}\right){}{{F}}_{{t}{,}{t}}^{{2}}{+}\left({12}{}{{c}}_{{1}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}}^{{4}}{}{u}{}\left({t}\right){+}{16}{}{{c}}_{{2}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}}{}{{F}}_{{t}}^{{3}}{+}{12}{}{{c}}_{{3}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{2}}\right){}{{F}}_{{t}{,}{t}}{+}{8}{}{{c}}_{{0}}{}\left({F}{}\left({t}\right)\right){}{{F}}_{{t}}^{{6}}{}{u}{}\left({t}\right){+}{8}{}{{c}}_{{1}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}}{}{{F}}_{{t}}^{{5}}{+}{8}{}{{c}}_{{2}}{}\left({F}{}\left({t}\right)\right){}{{u}}_{{t}{,}{t}}{}{{F}}_{{t}}^{{4}}}{{8}{}{{F}}_{{t}}^{{9}}{{2}}}}$ (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{{16}{}\left(\frac{{3}{}{u}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}}{{2}{}{{\mathrm{F\text{'}}}}^{{5}}{{2}}}}{+}\frac{\left({12}{}{{c}}_{{2}}{}\left({F}\right){}{{\mathrm{F\text{'}}}}^{{3}}{}{u}{-}{30}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}}}{}{u}{+}{28}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{u\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}\right){}{\mathrm{F\text{'}\text{'}\text{'}}}{+}{8}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{u\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{15}{}{{c}}_{{3}}{}\left({F}\right){}{{\mathrm{F\text{'}\text{'}}}}^{{3}}{}{u}{+}\left({-}{6}{}{{c}}_{{2}}{}\left({F}\right){}{{\mathrm{F\text{'}}}}^{{2}}{}{u}{-}{30}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}}}\right){}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{+}\left({12}{}{{c}}_{{1}}{}\left({F}\right){}{{\mathrm{F\text{'}}}}^{{4}}{}{u}{+}{16}{}{{c}}_{{2}}{}\left({F}\right){}{\mathrm{u\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{12}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{2}}\right){}{\mathrm{F\text{'}\text{'}}}{+}{8}{}{{c}}_{{0}}{}\left({F}\right){}{{\mathrm{F\text{'}}}}^{{6}}{}{u}{+}{8}{}{{c}}_{{1}}{}\left({F}\right){}{\mathrm{u\text{'}}}{}{{\mathrm{F\text{'}}}}^{{5}}{+}{8}{}{{c}}_{{2}}{}\left({F}\right){}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{4}}}{{8}{}{{\mathrm{F\text{'}}}}^{{9}}{{2}}}}\right){}{{\mathrm{F\text{'}}}}^{{13}}{{2}}}{-}{24}{}{u}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{-}{80}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{\mathrm{u\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{-}{80}{}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{}{{\mathrm{F\text{'}}}}^{{3}}{+}{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}}{+}{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}}{-}{300}{}{u}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{\mathrm{F\text{'}\text{'}\text{'}}}{}{\mathrm{F\text{'}}}{-}{240}{}{{\mathrm{F\text{'}\text{'}}}}^{{3}}{}{\mathrm{u\text{'}}}{}{\mathrm{F\text{'}}}{+}{135}{}{u}{}{{\mathrm{F\text{'}\text{'}}}}^{{4}}}{{16}{}{{\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}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}}{{2}}{-}\frac{{5}{}{\mathrm{F\text{'}\text{'}\text{'}}}}{{\mathrm{F\text{'}}}}{+}\frac{{15}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}}{{2}{}{{\mathrm{F\text{'}}}}^{{2}}}{,}{{C}}_{{1}}{=}{{\mathrm{F\text{'}}}}^{{3}}{}{{c}}_{{1}}{}\left({F}\right){+}{2}{}{\mathrm{F\text{'}}}{}{{c}}_{{2}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}{+}\frac{{7}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}\text{'}}}}{{2}}{-}\frac{{15}{}{{c}}_{{3}}{}\left({F}\right){}{{\mathrm{F\text{'}\text{'}}}}^{{2}}}{{4}{}{\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}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}}{{2}}{+}\frac{{3}{}{{\mathrm{F\text{'}}}}^{{2}}{}{\mathrm{F\text{'}\text{'}}}{}{{c}}_{{1}}{}\left({F}\right)}{{2}}{+}\frac{{3}{}{\mathrm{F\text{'}}}{}{{c}}_{{2}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}\text{'}}}}{{2}}{-}\frac{{3}{}{{c}}_{{2}}{}\left({F}\right){}{{\mathrm{F\text{'}\text{'}}}}^{{2}}}{{4}}{-}\frac{{15}{}{{c}}_{{3}}{}\left({F}\right){}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}}}}{{4}{}{\mathrm{F\text{'}}}}{+}\frac{{15}{}{{c}}_{{3}}{}\left({F}\right){}{{\mathrm{F\text{'}\text{'}}}}^{{3}}}{{8}{}{{\mathrm{F\text{'}}}}^{{2}}}{+}\frac{{15}{}{\mathrm{F\text{'}\text{'}}}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}}}}{{2}{}{{\mathrm{F\text{'}}}}^{{2}}}{-}\frac{{3}{}{\mathrm{F\text{'}\text{'}\text{'}\text{'}\text{'}}}}{{2}{}{\mathrm{F\text{'}}}}{+}\frac{{15}{}{{\mathrm{F\text{'}\text{'}\text{'}}}}^{{2}}}{{4}{}{{\mathrm{F\text{'}}}}^{{2}}}{-}\frac{{75}{}{{\mathrm{F\text{'}\text{'}}}}^{{2}}{}{\mathrm{F\text{'}\text{'}\text{'}}}}{{4}{}{{\mathrm{F\text{'}}}}^{{3}}}{+}\frac{{135}{}{{\mathrm{F\text{'}\text{'}}}}^{{4}}}{{16}{}{{\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

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

It is visible that each expression is now of the form ${\left(F\text{'}\right)}^{k}\mathrm{\Phi }\left({c}_{j}\left(F\right)\right)$, 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

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

Let's now transform the linear equation ode 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)
 > $\mathrm{ODEInvariants}\left(\mathrm{nonlinearODE}\right)$
 $\left[{{c}}_{{1}}{+}\frac{{{c}}_{{2}}{}{{c}}_{{3}}}{{2}}{+}\frac{{{c}}_{{3}}^{{3}}}{{8}}{+}\frac{{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}}{{2}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{-}\frac{{3}{}{{c}}_{{3}}{}{{c}}_{{3}}^{{\mathrm{\text{'}}}}}{{4}}{,}\frac{{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}\text{'}}}}}{{4}}{-}{{c}}_{{2}}^{{\mathrm{\text{'}\text{'}}}}{-}\frac{{3}{}{{c}}_{{3}}^{{\mathrm{\text{'}\text{'}}}}{}{{c}}_{{3}}}{{4}}{-}\frac{{33}{}{\left({{c}}_{{3}}^{{\mathrm{\text{'}}}}\right)}^{{2}}}{{40}}{+}\frac{\left({312}{}{{c}}_{{3}}^{{2}}{+}{432}{}{{c}}_{{2}}\right){}{{c}}_{{3}}^{{\mathrm{\text{'}}}}}{{320}}{-}\frac{{39}{}{{c}}_{{3}}^{{4}}}{{320}}{-}\frac{{13}{}{{c}}_{{3}}^{{2}}{}{{c}}_{{2}}}{{20}}{-}\frac{{5}{}{{c}}_{{1}}{}{{c}}_{{3}}}{{4}}{-}\frac{{9}{}{{c}}_{{2}}^{{2}}}{{20}}{+}\frac{{5}{}{{c}}_{{2}}^{{\mathrm{\text{'}}}}{}{{c}}_{{3}}}{{4}}{-}{5}{}{{c}}_{{0}}{+}\frac{{5}{}{{c}}_{{1}}^{{\mathrm{\text{'}}}}}{{2}}\right]$ (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) 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. 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

  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.