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 ${\mathrm{I}}_r$ and ${\mathrm{I}}_s$ respectively of weights $r$ and $s$, an absolute invariant can be constructed by taking $\frac{{\mathrm{I}}_r^s}{{\mathrm{I}}_s^r}$ (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,

by transforming this equation using

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.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{ODEInvariants}\right)$

$\left[\mathrm{ODEInvariants}\right]$

$\mathrm{PDEtools}:-\mathrm{declare}\left(\left(C\,F\,c\,u\,y\right)\left(x\right)\,\mathrm{prime}=x\right)$

$C\left(x\right)\mathrm{will\; now\; be\; displayed\; as}C$

$F\left(x\right)\mathrm{will\; now\; be\; displayed\; as}F$

$c\left(x\right)\mathrm{will\; now\; be\; displayed\; as}c$

$u\left(x\right)\mathrm{will\; now\; be\; displayed\; as}u$

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

$\mathrm{ode}\left[3\right]≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)=\mathrm{add}\left(c\left[j\right]\left(x\right)\mathrm{diff}\left(y\left(x\right)\,\left[\mathrm{`\$`}\left(x\,j\right)\right]\right)\,j=\left[2\,1\,0\right]\right)$

${\mathrm{ode}}_3≔\mathrm{y\text{'}\text{'}\text{'}}=c_2\mathrm{y\text{'}\text{'}}+c_1\mathrm{y\text{'}}+c_{0}y$

$\mathrm{ODEInvariants}\left(\mathrm{ode}\left[3\right]\right)$

$\left[2c_0+\frac{2c_1{c}_2}{3}+\frac{4{c_2}^3}{27}+\frac{c_2^{\mathrm{\text{'}\text{'}}}}{3}-c_1^{\text{'}}-\frac{2c_2^{\text{'}}{c}_2}{3}\right]$

$\mathrm{ode}\left[4\right]≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,4\right)\right)=\mathrm{add}\left(c\left[j\right]\left(x\right)\mathrm{diff}\left(y\left(x\right)\,\left[\mathrm{`\$`}\left(x\,j\right)\right]\right)\,j=\left[3\,2\,1\,0\right]\right)$

${\mathrm{ode}}_4≔\mathrm{y\text{'}\text{'}\text{'}\text{'}}=c_3\mathrm{y\text{'}\text{'}\text{'}}+c_2\mathrm{y\text{'}\text{'}}+c_1\mathrm{y\text{'}}+c_{0}y$

$\mathrm{ii}≔\mathrm{ODEInvariants}\left(\mathrm{ode}\left[4\right]\right)$

$\mathrm{ii}≔\left[c_1+\frac{c_2{c}_3}{2}+\frac{{c_3}^3}{8}+\frac{c_3^{\mathrm{\text{'}\text{'}}}}{2}-c_2^{\text{'}}-\frac{3c_3{c}_3^{\text{'}}}{4}\,\frac{c_3^{\mathrm{\text{'}\text{'}\text{'}}}}{4}-c_2^{\mathrm{\text{'}\text{'}}}-\frac{3c_3^{\mathrm{\text{'}\text{'}}}{c}_3}{4}-\frac{33{\left(c_3^{\text{'}}\right)}^2}{40}+\frac{3\left(13{c_3}^2+18c_2\right)c_3^{\text{'}}}{40}-\frac{39{c_3}^4}{320}-\frac{13{c_3}^2{c}_2}{20}-\frac{5c_1{c}_3}{4}-\frac{9{c_2}^2}{20}+\frac{5c_2^{\text{'}}{c}_3}{4}-5c_0+\frac{5c_1^{\text{'}}}{2}\right]$

$\mathrm{tr}≔\left\{x=F\left(t\right)\,y\left(x\right)={\mathrm{diff}\left(F\left(t\right)\,t\right)}^{\frac{3}{2}}u\left(t\right)\right\}$

$\mathrm{tr}≔\left\{x=F\left(t\right)\,y={F_t}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}u\left(t\right)\right\}$

$\mathrm{PDEtools}:-\mathrm{dchange}\left(\mathrm{tr}\,\mathrm{ode}\left[4\right]\,\left[t\,u\left(t\right)\right]\,\mathrm{known}=\mathrm{all}\,\mathrm{simplify}\right)$

$\frac{16u_{t,t,t,t}{F_t}^4+24u\left(t\right)F_{t,t,t,t,t}{F_t}^3+80F_{t,t,t}{u}_{t,t}{F_t}^3+80u_t{F}_{t,t,t,t}{F_t}^3-120u\left(t\right)F_{t,t}{F}_{t,t,t,t}{F_t}^2-60u\left(t\right){F_{t,t,t}}^2{F_t}^2-120{F_{t,t}}^2{u}_{t,t}{F_t}^2-320F_{t,t}{u}_t{F}_{t,t,t}{F_t}^2+300u\left(t\right){F_{t,t}}^2{F}_{t,t,t}{F}_t+240{F_{t,t}}^3{u}_t{F}_t-135u\left(t\right){F_{t,t}}^4}{16{F_t}^{\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=\frac{3c_3\left(F\left(t\right)\right)u\left(t\right)F_{t,t,t,t}{F_t}^2+\frac{\left(6c_2\left(F\left(t\right)\right)u\left(t\right){F_t}^3-15c_3\left(F\left(t\right)\right)u\left(t\right)F_{t,t}{F}_t+14c_3\left(F\left(t\right)\right)u_t{F_t}^2\right)F_{t,t,t}}{2}+2c_3\left(F\left(t\right)\right)u_{t,t,t}{F_t}^3+\frac{15c_3\left(F\left(t\right)\right)u\left(t\right){F_{t,t}}^3}{4}+\frac{3\left(-c_2\left(F\left(t\right)\right)u\left(t\right){F_t}^2-5c_3\left(F\left(t\right)\right)u_t{F}_t\right){F_{t,t}}^2}{2}+\left(3c_1\left(F\left(t\right)\right)u\left(t\right){F_t}^4+4c_2\left(F\left(t\right)\right)u_t{F_t}^3+3c_3\left(F\left(t\right)\right)u_{t,t}{F_t}^2\right)F_{t,t}+2c_0\left(F\left(t\right)\right)u\left(t\right){F_t}^6+2c_1\left(F\left(t\right)\right)u_t{F_t}^5+2c_2\left(F\left(t\right)\right)u_{t,t}{F_t}^4}{2{F_t}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{subs}\left(t=x\,\mathrm{isolate}\left(\,\mathrm{diff}\left(u\left(t\right)\,\mathrm{`\$`}\left(t\,4\right)\right)\right)\right)$

$\mathrm{u\text{'}\text{'}\text{'}\text{'}}=\frac{8{\mathrm{F\text{'}}}^2\left(3c_3\left(F\right)u\mathrm{F\text{'}\text{'}\text{'}\text{'}}{\mathrm{F\text{'}}}^2+\frac{\left(6c_2\left(F\right)u{\mathrm{F\text{'}}}^3-15c_3\left(F\right)u\mathrm{F\text{'}\text{'}}\mathrm{F\text{'}}+14c_3\left(F\right)\mathrm{u\text{'}}{\mathrm{F\text{'}}}^2\right)\mathrm{F\text{'}\text{'}\text{'}}}{2}+2c_3\left(F\right)\mathrm{u\text{'}\text{'}\text{'}}{\mathrm{F\text{'}}}^3+\frac{15c_3\left(F\right)u{\mathrm{F\text{'}\text{'}}}^3}{4}+\frac{3\left(-c_2\left(F\right)u{\mathrm{F\text{'}}}^2-5c_3\left(F\right)\mathrm{u\text{'}}\mathrm{F\text{'}}\right){\mathrm{F\text{'}\text{'}}}^2}{2}+\left(3c_1\left(F\right)u{\mathrm{F\text{'}}}^4+4c_2\left(F\right)\mathrm{u\text{'}}{\mathrm{F\text{'}}}^3+3c_3\left(F\right)\mathrm{u\text{'}\text{'}}{\mathrm{F\text{'}}}^2\right)\mathrm{F\text{'}\text{'}}+2c_0\left(F\right)u{\mathrm{F\text{'}}}^6+2c_1\left(F\right)\mathrm{u\text{'}}{\mathrm{F\text{'}}}^5+2c_2\left(F\right)\mathrm{u\text{'}\text{'}}{\mathrm{F\text{'}}}^4\right)-24u\mathrm{F\text{'}\text{'}\text{'}\text{'}\text{'}}{\mathrm{F\text{'}}}^3-80\mathrm{F\text{'}\text{'}\text{'}}\mathrm{u\text{'}\text{'}}{\mathrm{F\text{'}}}^3-80\mathrm{u\text{'}}\mathrm{F\text{'}\text{'}\text{'}\text{'}}{\mathrm{F\text{'}}}^3+120u\mathrm{F\text{'}\text{'}}\mathrm{F\text{'}\text{'}\text{'}\text{'}}{\mathrm{F\text{'}}}^2+60u{\mathrm{F\text{'}\text{'}\text{'}}}^2{\mathrm{F\text{'}}}^2+120{\mathrm{F\text{'}\text{'}}}^2\mathrm{u\text{'}\text{'}}{\mathrm{F\text{'}}}^2+320\mathrm{F\text{'}\text{'}}\mathrm{u\text{'}}\mathrm{F\text{'}\text{'}\text{'}}{\mathrm{F\text{'}}}^2-300u{\mathrm{F\text{'}\text{'}}}^2\mathrm{F\text{'}\text{'}\text{'}}\mathrm{F\text{'}}-240{\mathrm{F\text{'}\text{'}}}^3\mathrm{u\text{'}}\mathrm{F\text{'}}+135u{\mathrm{F\text{'}\text{'}}}^4}{16{\mathrm{F\text{'}}}^4}$

$\mathrm{zip}\left(\mathrm{`=`}\,\left[C\left[3\right]\left(x\right)\,C\left[2\right]\left(x\right)\,C\left[1\right]\left(x\right)\,C\left[0\right]\left(x\right)\right]\,\left[\mathrm{PDEtools}:-\mathrm{dcoeffs}\left(\mathrm{rhs}\left(\right)\,u\left(x\right)\right)\right]\right)$

$\left[C_3=\mathrm{F\text{'}}{c}_3\left(F\right)\,C_2={\mathrm{F\text{'}}}^2{c}_2\left(F\right)+\frac{3c_3\left(F\right)\mathrm{F\text{'}\text{'}}}{2}-\frac{5\mathrm{F\text{'}\text{'}\text{'}}}{\mathrm{F\text{'}}}+\frac{15{\mathrm{F\text{'}\text{'}}}^2}{2{\mathrm{F\text{'}}}^2}\,C_1={\mathrm{F\text{'}}}^3{c}_1\left(F\right)+2\mathrm{F\text{'}}{c}_2\left(F\right)\mathrm{F\text{'}\text{'}}+\frac{7c_3\left(F\right)\mathrm{F\text{'}\text{'}\text{'}}}{2}-\frac{15c_3\left(F\right){\mathrm{F\text{'}\text{'}}}^2}{4\mathrm{F\text{'}}}-\frac{5\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{\mathrm{F\text{'}}}+\frac{20\mathrm{F\text{'}\text{'}}\mathrm{F\text{'}\text{'}\text{'}}}{{\mathrm{F\text{'}}}^2}-\frac{15{\mathrm{F\text{'}\text{'}}}^3}{{\mathrm{F\text{'}}}^3}\,C_0={\mathrm{F\text{'}}}^4{c}_0\left(F\right)+\frac{3{\mathrm{F\text{'}}}^2\mathrm{F\text{'}\text{'}}{c}_1\left(F\right)}{2}+\frac{3\mathrm{F\text{'}}{c}_2\left(F\right)\mathrm{F\text{'}\text{'}\text{'}}}{2}-\frac{3c_2\left(F\right){\mathrm{F\text{'}\text{'}}}^2}{4}+\frac{3c_3\left(F\right)\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{2}-\frac{15c_3\left(F\right)\mathrm{F\text{'}\text{'}}\mathrm{F\text{'}\text{'}\text{'}}}{4\mathrm{F\text{'}}}+\frac{15c_3\left(F\right){\mathrm{F\text{'}\text{'}}}^3}{8{\mathrm{F\text{'}}}^2}-\frac{3\mathrm{F\text{'}\text{'}\text{'}\text{'}\text{'}}}{2\mathrm{F\text{'}}}+\frac{15\mathrm{F\text{'}\text{'}}\mathrm{F\text{'}\text{'}\text{'}\text{'}}}{2{\mathrm{F\text{'}}}^2}+\frac{15{\mathrm{F\text{'}\text{'}\text{'}}}^2}{4{\mathrm{F\text{'}}}^2}-\frac{75{\mathrm{F\text{'}\text{'}}}^2\mathrm{F\text{'}\text{'}\text{'}}}{4{\mathrm{F\text{'}}}^3}+\frac{135{\mathrm{F\text{'}\text{'}}}^4}{16{\mathrm{F\text{'}}}^4}\right]$

$\mathrm{subs}\left(c=C\,\mathrm{ii}\right)$

$\left[C_1+\frac{C_2{C}_3}{2}+\frac{{C_3}^3}{8}+\frac{C_3^{\mathrm{\text{'}\text{'}}}}{2}-C_2^{\text{'}}-\frac{3C_3{C}_3^{\text{'}}}{4}\,\frac{C_3^{\mathrm{\text{'}\text{'}\text{'}}}}{4}-C_2^{\mathrm{\text{'}\text{'}}}-\frac{3C_3^{\mathrm{\text{'}\text{'}}}{C}_3}{4}-\frac{33{\left(C_3^{\text{'}}\right)}^2}{40}+\frac{3\left(13{C_3}^2+18C_2\right)C_3^{\text{'}}}{40}-\frac{39{C_3}^4}{320}-\frac{13{C_3}^2{C}_2}{20}-\frac{5C_1{C}_3}{4}-\frac{9{C_2}^2}{20}+\frac{5C_2^{\text{'}}{C}_3}{4}-5C_0+\frac{5C_1^{\text{'}}}{2}\right]$

$\mathrm{factor}\left(\mathrm{eval}\left(\,\right)\right)$

$\left[\frac{{\mathrm{F\text{'}}}^3\left({c_3\left(F\right)}^3+4c_3\left(F\right)c_2\left(F\right)-6c_3\left(F\right)\mathrm{D}\left(c_3\right)\left(F\right)+8c_1\left(F\right)+4{\mathrm{D}}^{\left(2\right)}\left(c_3\right)\left(F\right)-8\mathrm{D}\left(c_2\right)\left(F\right)\right)}{8}\,\frac{{\mathrm{F\text{'}}}^4\left(-39{c_3\left(F\right)}^4-208{c_3\left(F\right)}^2{c}_2\left(F\right)+312{c_3\left(F\right)}^2\mathrm{D}\left(c_3\right)\left(F\right)-400c_1\left(F\right)c_3\left(F\right)-240c_3\left(F\right){\mathrm{D}}^{\left(2\right)}\left(c_3\right)\left(F\right)+400c_3\left(F\right)\mathrm{D}\left(c_2\right)\left(F\right)-144{c_2\left(F\right)}^2+432c_2\left(F\right)\mathrm{D}\left(c_3\right)\left(F\right)-264{\mathrm{D}\left(c_3\right)\left(F\right)}^2+800\mathrm{D}\left(c_1\right)\left(F\right)+80{\mathrm{D}}^{\left(3\right)}\left(c_3\right)\left(F\right)-320{\mathrm{D}}^{\left(2\right)}\left(c_2\right)\left(F\right)-1600c_0\left(F\right)\right)}{320}\right]$

$\mathrm{convert}\left(\mathrm{eval}\left(\,\mathrm{`=`}\left(F\,x\mapsto x\right)\right)\,\mathrm{diff}\right)$

$\left[c_1+\frac{c_2{c}_3}{2}+\frac{{c_3}^3}{8}+\frac{c_3^{\mathrm{\text{'}\text{'}}}}{2}-c_2^{\text{'}}-\frac{3c_3{c}_3^{\text{'}}}{4}\,-\frac{39{c_3}^4}{320}-\frac{13{c_3}^2{c}_2}{20}+\frac{39c_3^{\text{'}}{c_3}^2}{40}-\frac{5c_1{c}_3}{4}-\frac{3c_3^{\mathrm{\text{'}\text{'}}}{c}_3}{4}+\frac{5c_2^{\text{'}}{c}_3}{4}-\frac{9{c_2}^2}{20}+\frac{27c_2{c}_3^{\text{'}}}{20}-\frac{33{\left(c_3^{\text{'}}\right)}^2}{40}+\frac{5c_1^{\text{'}}}{2}+\frac{c_3^{\mathrm{\text{'}\text{'}\text{'}}}}{4}-c_2^{\mathrm{\text{'}\text{'}}}-5c_0\right]$

$\mathrm{normal}\left(-\mathrm{ii}\right)$

$\left[0\,0\right]$

$\mathrm{PDEtools}:-\mathrm{dchange}\left(y\left(x\right)=\frac{1}{u\left(x\right)}\,\mathrm{ode}\left[4\right]\,\left[u\left(x\right)\right]\right)$

$\frac{24{\mathrm{u\text{'}}}^4}{u^5}-\frac{36{\mathrm{u\text{'}}}^2\mathrm{u\text{'}\text{'}}}{u^4}+\frac{6{\mathrm{u\text{'}\text{'}}}^2}{u^3}+\frac{8\mathrm{u\text{'}}\mathrm{u\text{'}\text{'}\text{'}}}{u^3}-\frac{\mathrm{u\text{'}\text{'}\text{'}\text{'}}}{u^2}=c_3\left(-\frac{6{\mathrm{u\text{'}}}^3}{u^4}+\frac{6\mathrm{u\text{'}}\mathrm{u\text{'}\text{'}}}{u^3}-\frac{\mathrm{u\text{'}\text{'}\text{'}}}{u^2}\right)+c_2\left(\frac{2{\mathrm{u\text{'}}}^2}{u^3}-\frac{\mathrm{u\text{'}\text{'}}}{u^2}\right)-\frac{c_1\mathrm{u\text{'}}}{u^2}+\frac{c_0}{u}$

$\mathrm{nonlinearODE}≔\mathrm{isolate}\left(\,\mathrm{diff}\left(u\left(x\right)\,\mathrm{`\$`}\left(x\,4\right)\right)\right)$

$\mathrm{nonlinearODE}≔\mathrm{u\text{'}\text{'}\text{'}\text{'}}=-\left(c_3\left(-\frac{6{\mathrm{u\text{'}}}^3}{u^4}+\frac{6\mathrm{u\text{'}}\mathrm{u\text{'}\text{'}}}{u^3}-\frac{\mathrm{u\text{'}\text{'}\text{'}}}{u^2}\right)+c_2\left(\frac{2{\mathrm{u\text{'}}}^2}{u^3}-\frac{\mathrm{u\text{'}\text{'}}}{u^2}\right)-\frac{c_1\mathrm{u\text{'}}}{u^2}+\frac{c_0}{u}-\frac{24{\mathrm{u\text{'}}}^4}{u^5}+\frac{36{\mathrm{u\text{'}}}^2\mathrm{u\text{'}\text{'}}}{u^4}-\frac{6{\mathrm{u\text{'}\text{'}}}^2}{u^3}-\frac{8\mathrm{u\text{'}}\mathrm{u\text{'}\text{'}\text{'}}}{u^3}\right)u^2$

$\mathrm{ODEInvariants}\left(\mathrm{nonlinearODE}\right)$

$\left[c_1+\frac{c_2{c}_3}{2}+\frac{{c_3}^3}{8}+\frac{c_3^{\mathrm{\text{'}\text{'}}}}{2}-c_2^{\text{'}}-\frac{3c_3{c}_3^{\text{'}}}{4}\,\frac{c_3^{\mathrm{\text{'}\text{'}\text{'}}}}{4}-c_2^{\mathrm{\text{'}\text{'}}}-\frac{3c_3^{\mathrm{\text{'}\text{'}}}{c}_3}{4}-\frac{33{\left(c_3^{\text{'}}\right)}^2}{40}+\frac{3\left(13{c_3}^2+18c_2\right)c_3^{\text{'}}}{40}-\frac{39{c_3}^4}{320}-\frac{13{c_3}^2{c}_2}{20}-\frac{5c_1{c}_3}{4}-\frac{9{c_2}^2}{20}+\frac{5c_2^{\text{'}}{c}_3}{4}-5c_0+\frac{5c_1^{\text{'}}}{2}\right]$

$\mathrm{diff}\left(u\left(x\right)\,\mathrm{`\$`}\left(x\,4\right)\right)={u\left(x\right)}^3\mathrm{diff}\left(u\left(x\right)\,x\right)+{\mathrm{diff}\left(u\left(x\right)\,x\,x\right)}^2+x$

$\mathrm{u\text{'}\text{'}\text{'}\text{'}}=u^3\mathrm{u\text{'}}+{\mathrm{u\text{'}\text{'}}}^2+x$

$\mathrm{ODEInvariants}\left(\right)$

$\left[u^3-2\mathrm{u\text{'}\text{'}\text{'}}\,-\frac{15\mathrm{u\text{'}}{u}^2}{2}-\frac{19{\mathrm{u\text{'}\text{'}}}^2}{5}-2u^3\mathrm{u\text{'}}-2x\right]$

[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.