The following example is from [HW96]. It features a differential-algebraic system of differential index $2$, describing a steering wheel. The dependent variables $x\left(t\right)$ and $y\left(t\right)$ are space coordinates. The dependent variable $z\left(t\right)$ is a command and the system is given by

To compute a complete set of relations, defining the algebraic variety, over which the integral curves of the system must remain. These relations are essential for choosing consistent initial values.

To compute the missing ODE, which expresses the evolution of $z\left(t\right)$.

$\mathrm{sys}\left[1\right]≔\left[\mathrm{diff}\left(x\left(t\right)\,t\right)=\frac{7}{10}y\left(t\right)+s\left(t\right)\,\mathrm{diff}\left(y\left(t\right)\,t\right)=\frac{14}{10}x\left(t\right)+c\left(t\right)\,{x\left(t\right)}^2+{y\left(t\right)}^2=1\,\mathrm{diff}\left(s\left(t\right)\,t\right)=\frac{25}{10}\mathrm{diff}\left(z\left(t\right)\,t\right)c\left(t\right)\,\mathrm{diff}\left(c\left(t\right)\,t\right)=-\frac{25}{10}\mathrm{diff}\left(z\left(t\right)\,t\right)s\left(t\right)\,{s\left(t\right)}^2+{c\left(t\right)}^2=1\right]$

${\mathrm{sys}}_1≔\left[\frac{\ⅆ}{\ⅆt}x\left(t\right)=\frac{7y\left(t\right)}{10}+s\left(t\right)\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=\frac{7x\left(t\right)}{5}+c\left(t\right)\,{x\left(t\right)}^2+{y\left(t\right)}^2=1\,\frac{\ⅆ}{\ⅆt}s\left(t\right)=\frac{5\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)c\left(t\right)}{2}\,\frac{\ⅆ}{\ⅆt}c\left(t\right)=-\frac{5\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)s\left(t\right)}{2}\,{s\left(t\right)}^2+{c\left(t\right)}^2=1\right]$

Then, a differential polynomial ring is defined, endowed with an orderly ranking, and the system is simplified.

$\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right)\:$

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{blocks}=\left[\left[z\,y\,x\,s\,c\right]\right]\right)$

$R≔\mathrm{differential\_ring}$

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\mathrm{sys}\left[1\right]\,R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\left[1\right]\,\mathrm{order}=0\,\mathrm{notation}=\mathrm{jet}\,\mathrm{solved}\right)$

$\left[y=-\frac{100c^{2}sx-210cs{x}^2-441s{x}^3+210cs+341sx}{100c^3-100c}\,x^4=-\frac{20}{21}{x}^{3}c+\frac{341}{441}{x}^2+\frac{20}{21}xc+\frac{100}{441}{c}^2\,s^2=-c^2+1\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\left[1\right]\,\mathrm{leader}=\mathrm{derivative}\left(z\left(t\right)\right)\,\mathrm{notation}=\mathrm{jet}\,\mathrm{solved}\right)$

$\left[z_t=-\frac{55566000c^5{x}^2+58344300c^4{x}^3-37044000c^5-43306200c^{4}x-56831670c^3{x}^2-54519507c^2{x}^3+36335670c^3+40951407c^{2}x+13461000c{x}^2+10892700x^3-7987000c-8422700x}{11025000c^5-11025000c^3+2500000c}\right]$

This example features a model reduction for chemical reaction systems, based on a quasi-steady state approximation [BLLM07]. The process is illustrated over a famous case, but has more general applications.

$\mathrm{sys}\left[2\right]≔\left[\mathrm{diff}\left(E\left(t\right)\,t\right)=-k\left[1\right]E\left(t\right)S\left(t\right)+k\left[-1\right]C\left(t\right)+k\left[2\right]C\left(t\right)\,\mathrm{diff}\left(S\left(t\right)\,t\right)=-k\left[1\right]E\left(t\right)S\left(t\right)+k\left[-1\right]C\left(t\right)\,\mathrm{diff}\left(C\left(t\right)\,t\right)=k\left[1\right]E\left(t\right)S\left(t\right)-k\left[-1\right]C\left(t\right)-k\left[2\right]C\left(t\right)\,\mathrm{diff}\left(P\left(t\right)\,t\right)=k\left[2\right]C\left(t\right)\right]$

${\mathrm{sys}}_2≔\left[\frac{\ⅆ}{\ⅆt}E\left(t\right)=-k_{1}E\left(t\right)S\left(t\right)+k_{\mathrm{-1}}C\left(t\right)+k_{2}C\left(t\right)\,\frac{\ⅆ}{\ⅆt}S\left(t\right)=-k_{1}E\left(t\right)S\left(t\right)+k_{\mathrm{-1}}C\left(t\right)\,\frac{\ⅆ}{\ⅆt}C\left(t\right)=k_{1}E\left(t\right)S\left(t\right)-k_{\mathrm{-1}}C\left(t\right)-k_{2}C\left(t\right)\,\frac{\ⅆ}{\ⅆt}P\left(t\right)=k_{2}C\left(t\right)\right]$

First, in the system, replace the contribution $k_{1}E\left(t\right)S\left(t\right)-k_{\mathrm{-1}}C\left(t\right)$ of the fast reactions, which occurs in the first three ODEs, by a new dependent variable $F_1$. This variable represents the unknown contribution, which cannot be assumed to be $0$, of the fast reactions, after the transient step. Somehow, this variable simply encodes that fast reactions are reactions.

Second, add the algebraic equation $k_{1}E\left(t\right)S\left(t\right)-k_{\mathrm{-1}}C\left(t\right)=0$ to the system. This equation forces the dynamics of the system to stays over the algebraic variety that it defines. Somehow, it encodes the speed.

Third, eliminate the new dependent variable $F_1$ from the differential-algebraic system. This elimination, performed using RosenfeldGroebner, computes the simplest ODE which describes the evolution of $S\left(t\right)$, and which is a consequence of the DAE.

$\mathrm{DAE}≔\left[\mathrm{diff}\left(E\left(t\right)\,t\right)=-F\left[1\right]\left(t\right)+k\left[2\right]C\left(t\right)\,\mathrm{diff}\left(S\left(t\right)\,t\right)=-F\left[1\right]\left(t\right)\,\mathrm{diff}\left(C\left(t\right)\,t\right)=F\left[1\right]\left(t\right)-k\left[2\right]C\left(t\right)\,\mathrm{diff}\left(P\left(t\right)\,t\right)=k\left[2\right]C\left(t\right)\,k\left[1\right]E\left(t\right)S\left(t\right)-k\left[-1\right]C\left(t\right)=0\right]$

$\mathrm{DAE}≔\left[\frac{\ⅆ}{\ⅆt}E\left(t\right)=-F_1\left(t\right)+k_{2}C\left(t\right)\,\frac{\ⅆ}{\ⅆt}S\left(t\right)=-F_1\left(t\right)\,\frac{\ⅆ}{\ⅆt}C\left(t\right)=F_1\left(t\right)-k_{2}C\left(t\right)\,\frac{\ⅆ}{\ⅆt}P\left(t\right)=k_{2}C\left(t\right)\,k_{1}E\left(t\right)S\left(t\right)-k_{\mathrm{-1}}C\left(t\right)=0\right]$

$\mathrm{species}≔\left[C\,E\,P\,S\right]$

$\mathrm{species}≔\left[C\,E\,P\,S\right]$

$\mathrm{rate\_constants}≔\left[k\left[1\right]\,k\left[-1\right]\,k\left[2\right]\right]$

$\mathrm{rate\_constants}≔\left[k_1\,k_{\mathrm{-1}}\,k_2\right]$

$R≔\mathrm{DifferentialRing}\left(\mathrm{blocks}=\left[F\left[1\right]\,\mathrm{species}\right]\,\mathrm{derivations}=\left[t\right]\,\mathrm{arbitrary}=\mathrm{rate\_constants}\right)$

$R≔\mathrm{differential\_ring}$

The third step can now be performed. An inequation,  $C\left(t\right)\ne 0$, is added to the $\mathrm{DAE}$ system passed to RosenfeldGroebner to avoid a spurious discussion on the case $C\left(t\right)=0$.

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[\mathrm{op}\left(\mathrm{DAE}\right)\,C\left(t\right)\ne 0\right]\,R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\right]$

The returned regular differential chain involves a differential polynomial expressing the evolution of $S\left(t\right)$; however, this relation is not yet the Henri-Michaelis-Menten formula. Indeed, the minor hypotheses have not yet been taken into account. This formula is just the simplest formula that can be deduced from the DAE, by using the main assumption only.

$\mathrm{Equations}\left(\mathrm{ideal}\left[1\right]\,\mathrm{leader}=\mathrm{diff}\left(S\left(t\right)\,t\right)\,\mathrm{solved}\right)$

$\left[\frac{\ⅆ}{\ⅆt}S\left(t\right)=-\frac{E\left(t\right){S\left(t\right)}^2{k}_1^2{k}_2+E\left(t\right)S\left(t\right)k_1{k}_{\mathrm{-1}}{k}_2}{E\left(t\right)k_1{k}_{\mathrm{-1}}+S\left(t\right)k_1{k}_{\mathrm{-1}}+k_{\mathrm{-1}}^2}\right]$

A few more parameters, $\mathrm{C0},\mathrm{E0},\mathrm{P0}$ and $\mathrm{S0}$, are introduced and the linear conservation laws of the system involving them:

$\mathrm{conservation\_laws}≔\left[E\left(t\right)+C\left(t\right)=\mathrm{E0}+\mathrm{C0}\,S\left(t\right)+C\left(t\right)+P\left(t\right)=\mathrm{S0}+\mathrm{C0}+\mathrm{P0}\right]$

$\mathrm{conservation\_laws}≔\left[E\left(t\right)+C\left(t\right)=\mathrm{E0}+\mathrm{C0}\,S\left(t\right)+C\left(t\right)+P\left(t\right)=\mathrm{S0}+\mathrm{C0}+\mathrm{P0}\right]$

A minor assumption is: $P\left(0\right)$ = $C\left(0\right)$ = $0$.

$\mathrm{hypotheses}≔\mathrm{subs}\left(\mathrm{P0}=0\,\mathrm{C0}=0\,\mathrm{conservation\_laws}\right)$

$\mathrm{hypotheses}≔\left[E\left(t\right)+C\left(t\right)=\mathrm{E0}\,S\left(t\right)+C\left(t\right)+P\left(t\right)=\mathrm{S0}\right]$

One could actually proceed from the formula established above, but, it is simpler to perform the whole simplification anew.

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[\mathrm{op}\left(\mathrm{DAE}\right)\,\mathrm{op}\left(\mathrm{hypotheses}\right)\right]\,R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\right]$

$\mathrm{formula}≔\mathrm{Equations}\left(\mathrm{ideal}\left[1\right]\,\mathrm{leader}=\mathrm{diff}\left(S\left(t\right)\,t\right)\,\mathrm{solved}\right)$

$\mathrm{formula}≔\left[\frac{\ⅆ}{\ⅆt}S\left(t\right)=-\frac{{S\left(t\right)}^2{k}_1^2{k}_2\mathrm{E0}+S\left(t\right)k_1{k}_{\mathrm{-1}}{k}_2\mathrm{E0}}{{S\left(t\right)}^2{k}_1^2+2S\left(t\right)k_1{k}_{\mathrm{-1}}+k_1{k}_{\mathrm{-1}}\mathrm{E0}+k_{\mathrm{-1}}^2}\right]$

Not yet: one still needs to perform some renaming on parameters, and to neglect the term $K\mathrm{E0}$, assuming $S\left(0\right)$ >> $E\left(0\right)$.

$\mathrm{formula}≔\mathrm{eval}\left(\mathrm{formula}\,k\left[-1\right]=Kk\left[1\right]\right)\:$

$\mathrm{formula}≔\mathrm{normal}\left(\mathrm{formula}\right)\:$

$\mathrm{formula}≔\mathrm{algsubs}\left(k\left[2\right]\mathrm{E0}=V\left[\max \right]\,\mathrm{formula}\right)\:$

Here is the Henri-Michaelis-Menten formula.

$\mathrm{formula}≔\mathrm{normal}\left(\mathrm{eval}\left(\mathrm{formula}\,K\mathrm{E0}=0\right)\right)$

$\mathrm{formula}≔\left[\frac{\ⅆ}{\ⅆt}S\left(t\right)=-\frac{V_{\max }S\left(t\right)}{K+S\left(t\right)}\right]$

Consider the set of differential polynomials describing the motion of a pendulum, that is, a point mass $m$ suspended from a massless rod of length $l$ under the influence of gravity $g$, in Cartesian coordinates $x,y$. The Lagrangian formulation leads to two second order differential equations with an algebraic constraint. $T$ is the Lagrangian multiplier.

$\mathrm{params}≔\left[m\,l\,g\right]\:$

$P≔\left[m\mathrm{diff}\left(y\left(t\right)\,t\,t\right)+T\left(t\right)y\left(t\right)+g\,m\mathrm{diff}\left(x\left(t\right)\,t\,t\right)+T\left(t\right)x\left(t\right)\,{x\left(t\right)}^2+{y\left(t\right)}^2-l^2\right]$

$P≔\left[m\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)+T\left(t\right)y\left(t\right)+g\,m\left(\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\right)+T\left(t\right)x\left(t\right)\,{x\left(t\right)}^2+{y\left(t\right)}^2-l^2\right]$

To find the lowest order differential polynomials vanishing on the zeros of the system modeling the pendulum and in particular all the algebraic constraints (that is, the differential polynomials of order $0$), use an $\mathrm{orderly}$ ranking. The second component in the output of RosenfeldGroebner corresponds to the equilibria of the pendulum. From the first component of the output, it can be seen that the actual motion is described by two first order differential equations with a constraint: the solution depends on only two arbitrary constants.

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{blocks}=\left[\left[T\,x\,y\right]\right]\,\mathrm{arbitrary}=\mathrm{params}\right)$

$R≔\mathrm{differential\_ring}$

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(P\,R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\,\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\left[1\right]\,\mathrm{notation}=\mathrm{jet}\,\mathrm{solved}\right)$

$\left[T_t=-\frac{3y_{t}g}{l^2}\,y_t^2=-\frac{-T{l}^4+T{l}^2{y}^2-g{l}^{2}y+g{y}^3}{m{l}^2}\,x^2=l^2-y^2\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\left[2\right]\,\mathrm{notation}=\mathrm{jet}\,\mathrm{solved}\right)$

$\left[T=-\frac{yg}{l^2}\,x=0\,y^2=l^2\right]$

To obtain the differential system satisfied by $x$ and $y$ alone, eliminate $T$. Note here that the motion is given by a second order differential equation in $y$. Then the Lagrange multiplier is given explicitly in terms of $y$ and its first derivative.

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right]\,\mathrm{blocks}=\left[T\,\left[x\,y\right]\right]\,\mathrm{arbitrary}=\mathrm{params}\right)$

$R≔\mathrm{differential\_ring}$

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(P\,R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\,\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\&comma;\mathrm{notation}=\mathrm{jet}\&comma;\mathrm{leader}<T\left(t\right)\&comma;\mathrm{solved}\right)$

$\left[\left[y_{t,t}=-\frac{-l^{2}my{y}_t^2-g{l}^4+2g{l}^2{y}^2-g{y}^4}{-l^{4}m+l^{2}m{y}^2}\&comma;x^2=l^2-y^2\right]\&comma;\left[x=0\&comma;y^2=l^2\right]\right]$

In general, when RosenfeldGroebner is applied on a single differential polynomial $F$, it returns at least one component involving $F$, and, possibly, some other components. The first component describes the general solution of $F$. The other ones describe the singular solutions.

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[t\right]\&comma;\mathrm{blocks}=\left[y\right]\right)$

$R≔\mathrm{differential\_ring}$

$F≔{\mathrm{diff}\left(y\left(t\right)\&comma;t\right)}^3-4ty\left(t\right)\mathrm{diff}\left(y\left(t\right)\&comma;t\right)+8{y\left(t\right)}^2$

$F≔{\left(\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right)}^3-4ty\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right)+8{y\left(t\right)}^2$

Over this example, RosenfeldGroebner returns the general component plus two singular ones. As we shall see, one of the singular components is essential, while the other one is not [K73,H99].

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\left[F\right]\&comma;R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\&comma;\mathrm{regular\_differential\_chain}\&comma;\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\right)$

$\left[\left[{\left(\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right)}^3-4ty\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right)+8{y\left(t\right)}^2\right]\&comma;\left[27y\left(t\right)-4t^3\right]\&comma;\left[y\left(t\right)\right]\right]$

The three components can be solved explicitly. The solutions of the general component writes $y\left(t\right)=c{\left(t-c\right)}^2$, where $c$ denotes an arbitrary constant. The solution of the first singular component writes $y\left(t\right)=\frac{4t^3}{27}$. It is not a particular case of the general solution. This first singular component is said to be essential. The solution of the second singular component writes $y\left(t\right)=0$. It is a particular case of the general solution, since it is obtained for $c=0$. This second singular component is said to be non-essential.

The function RosenfeldGroebner  with the option singsol = essential only returns the essential components of $F$:

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(F\&comma;R\&comma;\mathrm{singsol}=\mathrm{essential}\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\&comma;\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\right)$

$\left[\left[{\left(\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right)}^3-4ty\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right)+8{y\left(t\right)}^2\right]\&comma;\left[27y\left(t\right)-4t^3\right]\right]$

The regular differential chains returned by RosenfeldGroebner, permit to compute integral formal power series solutions of the input system. By integral, it is meant that the exponents are integers. By formal, it is meant that convergence issues are not addressed. The following system provides a good example of what a general PDE polynomial system may look like, Its power series solutions turn out to be polynomials.

$R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x\&comma;y\right]\&comma;\mathrm{blocks}=\left[\left[v\&comma;u\right]\right]\right)$

$R≔\mathrm{differential\_ring}$

$\mathrm{sys}\left[3\right]≔\left[{u\left[x\right]}^2-4u\&comma;u\left[x,y\right]v\left[y\right]-u+1\&comma;v\left[x,x\right]-u\left[x\right]\right]$

${\mathrm{sys}}_3≔\left[u_x^2-4u\&comma;u_{x,y}{v}_y-u+1\&comma;v_{x,x}-u_x\right]$

$\mathrm{ideal}≔\mathrm{RosenfeldGroebner}\left(\mathrm{sys}\left[3\right]\&comma;R\right)$

$\mathrm{ideal}≔\left[\mathrm{regular\_differential\_chain}\right]$

$\mathrm{Equations}\left(\mathrm{ideal}\left[1\right]\&comma;\mathrm{solved}\right)$

$\left[v_{x,x}=u_x\&comma;v_y=-\frac{-u{u}_x{u}_y+u_x{u}_y}{4u}\&comma;u_x^2=4u\&comma;u_y^2=2u\right]$

Looking at the leading derivatives of the returned regular differential chain, one sees that the system solutions depend on three constants, or initial values, which can be chosen almost arbitrary (see [DL84] for related undecidability results). Indeed, they must be solutions of the polynomial system of equations, returned by PowerSeriesSolution with the option conditions:

$\mathrm{PowerSeriesSolution}\left(\mathrm{conditions}\&comma;\mathrm{ideal}\left[1\right]\right)$

$\left[\left[{{\mathrm{D}}_1\left(u\right)\left(0\&comma;0\right)}^2-4u\left(0\&comma;0\right)=0\&comma;{{\mathrm{D}}_2\left(u\right)\left(0\&comma;0\right)}^2-2u\left(0\&comma;0\right)=0\right]\&comma;\left[u\left(0\&comma;0\right)\ne 0\&comma;{\mathrm{D}}_1\left(u\right)\left(0\&comma;0\right)\ne 0\&comma;{\mathrm{D}}_2\left(u\right)\left(0\&comma;0\right)\ne 0\right]\right]$

Let us choose the following initial values, and, check that they do satisfy the above polynomial system.

$\mathrm{iv}≔\left[u={c\left[0\right]}^2\&comma;u\left[y\right]=\mathrm{sqrt}\left(2\right)c\left[0\right]\&comma;u\left[x\right]=2c\left[0\right]\&comma;v=c\left[1\right]\&comma;v\left[x\right]=c\left[2\right]\right]$

$\mathrm{iv}≔\left[u=c_0^2\&comma;u_y=\sqrt{2}{c}_0\&comma;u_x=2c_0\&comma;v=c_1\&comma;v_x=c_2\right]$

$\mathrm{PowerSeriesSolution}\left(\mathrm{conditions}\&comma;\mathrm{ideal}\left[1\right]\&comma;\mathrm{iv}\right)$

$\left[\left[0=0\&comma;0=0\right]\&comma;\left[c_0\ne 0\right]\right]$

The PowerSeriesSolution function can now compute the integral formal power series solutions of ${\mathrm{sys}}_3$, for these initial values.

$\mathrm{sols}≔\mathrm{PowerSeriesSolution}\left(\mathrm{ideal}\left[1\right]\&comma;3\&comma;\mathrm{iv}\right)$

$\mathrm{sols}≔\left[v\left(x\&comma;y\right)=c_1+\left(\frac{c_0^2\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\right)y+c_{2}x+\frac{c_0{y}^2}{2}+\sqrt{2}{c}_{0}xy+c_0{x}^2+\frac{\sqrt{2}{y}^3}{12}+\frac{x{y}^2}{2}+\frac{\sqrt{2}{x}^{2}y}{2}+\frac{x^3}{3}\&comma;u\left(x\&comma;y\right)=c_0^2+\sqrt{2}{c}_{0}y+2c_{0}x+\frac{y^2}{2}+\sqrt{2}xy+x^2\right]$

$\mathrm{equations}≔\mathrm{Equations}\left(\mathrm{sys}\left[3\right]\&comma;R\&comma;\mathrm{notation}=\mathrm{diff}\right)$

$\mathrm{equations}≔\left[\frac{{\partial }^2}{\partial x^2}v\left(x\&comma;y\right)-\frac{\partial }{\partial x}u\left(x\&comma;y\right)\&comma;\left(\frac{{\partial }^2}{\partial x\partial y}u\left(x\&comma;y\right)\right)\left(\frac{\partial }{\partial y}v\left(x\&comma;y\right)\right)-u\left(x\&comma;y\right)+1\&comma;{\left(\frac{\partial }{\partial x}u\left(x\&comma;y\right)\right)}^2-4u\left(x\&comma;y\right)\right]$

$\mathrm{expand}\left(\mathrm{eval}\left(\mathrm{equations}\&comma;\mathrm{sols}\right)\right)$

$\left[0\&comma;0\&comma;0\right]$