The casesplit function receives one DE,  or a set or a list of DEs (it is also possible to include inequations and non-differential equations) and splits the system into regular systems associated with different cases. The union of the non-singular solutions of these regular systems returned is equal to the full set of solutions, including the singular ones, of the original input system. Each regular system automatically satisfies all the integrability conditions. The splitting into cases is performed by using the Rif subpackage of DEtools (default), or using the DifferentialAlgebra package when the keyword 'diffalg' is given as an optional argument. casesplit also works with anticommutative variables set using the Physics package using the approach explained in PerformOnAnticommutativeSystem.

The casesplit command is useful for studying the different cases of a given DE system, uncoupling the system in different ways using different orderings for the solving variables (rankings), and determining whether the system is inconsistent. The casesplit command can also be used to remove redundancies and to make the system complete, including simplifying it by using all its integrability conditions. All these tasks are automatically performed each time casesplit is invoked.

Ranking is a fundamental concept related to the splitting process performed through casesplit. A ranking is basically an ordering of the variables of the problem (herein called solving variables) in the form of a list of functions or symbol variables, as in $\left[f\left(x\right)\,g\left(x\right)\,h\left(x\right)\,a\right]$ (one can also specify functions by using only their names as in $\left[f\,g\,h\,a\right]$). Different rankings imply different uncoupling orderings and, hence, different manners of splitting and uncoupling the DE system (see checkrank, DifferentialAlgebra[Tools][Display], and the examples below).

In the case of a system of ordinary differential equations (ODEs), provided the system is not subdetermined, the default output of casesplit consists of a sequence of uncoupled ODE systems, where the uncoupling ordering used is that of the solving variables in the ranking. The ranking is provided by the user as a list or automatically set by casesplit. That means, for instance, that if a system depending on the functions $\left[f\left(x\right)\,g\left(x\right)\,h\left(x\right)\right]$ is given with this list as ranking, the equations in each returned system are of the form $\[\mathrm{..}f\left(x\right)\mathrm{..}=\mathrm{..}\left(f\left(x\right),g\left(x\right),h\left(x\right)\right)\mathrm{..},\mathrm{..}g\left(x\right)=\mathrm{..}\left(g\left(x\right),h\left(x\right)\right)\mathrm{..},\mathrm{..}h\left(x\right)\mathrm{..}=\mathrm{..}\left(h\left(x\right)\right)\]$. So, the last equation involves only $h\left(x\right)$, the next to the last equation involves $g\left(x\right)$ and $h\left(x\right)$, and the first equation involves the three functions. This is called sequential uncoupling.

In the case of a system of partial differential equations (PDEs), the situation is the same as with ODEs but the concept of sequential uncoupling requires clarification. Concretely, instead of one equation related to each function, one could receive many equations. For instance, if a system depending on the functions $\left[f\left(x\,y\,z\right)\,g\left(x\,y\,z\right)\,h\left(x\,y\,z\right)\right]$ is given and with this list as ranking, the equations in each returned system are of the form $\[\mathrm{eqs}\left(f=\mathrm{..}\left(f,g,h\right)\mathrm{..}\right),\mathrm{eqs}\left(g=\mathrm{..}\left(g,h\right)\mathrm{..}\right),\mathrm{eqs}\left(h=\mathrm{..}\left(h\right)\mathrm{..}\right)\]$, where $\mathrm{eqs}\left(h=\mathrm{..}\left(h\right)\mathrm{..}\right)$ means equations (possibly PDEs) only depending on 'h' and so on.

The casesplit command also works with systems of equations that involve mathematical functions, provided that they have a differential equation representation that is rational in the independent variables, the function and its derivatives - i.e a differential polynomial form. That is indeed the case of most of the mathematical functions as well as of indefinite integrals, fractional and symbolic (generic) powers. For example, the exponential function, $f\left(x\right)={\ⅇ}^x$ satisfies $f^{\text{'}}=f$. When these functions are present in the system, they are replaced by auxiliary unknowns representing them and the corresponding differential equations they satisfy are added to the system, the system is processed, and before returning these auxiliary unknowns are replaced by the actual functions they represent.

The arguments of mathematical functions present in the system can be arbitrary algebraic expressions, possibly including other mathematical functions, with no limits to the levels of nesting, for example: $g\left(x\right)={\ⅇ}^{\mathrm{BesselJ}\left(n\,\sqrt{x}\right)}$. These cases are processed by first performing changes of variables that result in a differential polynomial form representing the expression, then proceeding as explained in the previous item.

The casesplit command also works with systems that involve unknown functions whose arguments are not just variables (i.e.: of type name) but generic algebraic expressions. Provided that some conditions are met, through changes of variables the system of equations can be transformed into an equivalent system where the transformed functions all depend on simple variables of type name, this system is processed, a further change of variables to revert to the original variables is performed, and the result is returned.

Finally, casesplit works as well with systems that involve D derivatives evaluated at points represented by algebraic expressions. The key idea is that any D derivative of this type can be mapped into a function of variables not of type name that can in turn be represented in differential polynomial form as a first order PDE, obtained by running backwards what is know as the PDE characteristic strip solving method. This approach makes it possible to perform differential elimination with these objects, as well as solving PDE systems in terms of arbitrary functions, that, at some point in the solving process, become functions of algebraic expressions (variables not of type name) and where the intermediate subsystem involves D derivatives of these functions.

with(PDEtools):

with(DEtools):

declare( [f(t), g(t), h(t), j(t), y(t)], prime=t);

$f\left(t\right)\mathrm{will\; now\; be\; displayed\; as}f$

$g\left(t\right)\mathrm{will\; now\; be\; displayed\; as}g$

$h\left(t\right)\mathrm{will\; now\; be\; displayed\; as}h$

$j\left(t\right)\mathrm{will\; now\; be\; displayed\; as}j$

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

$\mathrm{derivatives\; with\; respect\; to}t\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

ode[1] := diff(y(t),t,t)^2 + 2*diff(y(t),t,t)*y(t)^3*diff(y(t),t) - 4*y(t)^2*diff(y(t),t)^3;

${\mathrm{ode}}_1≔{\mathrm{y\text{'}\text{'}}}^2+2\mathrm{y\text{'}\text{'}}{y}^3\mathrm{y\text{'}}-4y^2{\mathrm{y\text{'}}}^3$

[casesplit(ode[1])];

$\left[\left[{\mathrm{y\text{'}\text{'}}}^2=-2\mathrm{y\text{'}\text{'}}{y}^3\mathrm{y\text{'}}+4y^2{\mathrm{y\text{'}}}^3\right]\mathbf{where}\left[y^3\mathrm{y\text{'}}+\mathrm{y\text{'}\text{'}}\ne 0\right]\,\left[\mathrm{y\text{'}}=-\frac{y^4}{4}\right]\mathbf{where}\left[y\ne 0\right]\,\left[\mathrm{y\text{'}}=0\right]\mathbf{where}\left[y\ne 0\right]\,\left[y=0\right]\mathbf{where}\left[\right]\right]$

casesplit(ode[1], 'caseplot');

$\mathrm{=========\; Pivots\; Legend\; =========}$

$\mathrm{p1}=y^3\mathrm{y\text{'}}+\mathrm{y\text{'}\text{'}}$

$\mathrm{p2}=y$

$\mathrm{p3}=\mathrm{y\text{'}}$

$\left[{\mathrm{y\text{'}\text{'}}}^2=-2\mathrm{y\text{'}\text{'}}{y}^3\mathrm{y\text{'}}+4y^2{\mathrm{y\text{'}}}^3\right]\mathbf{where}\left[y^3\mathrm{y\text{'}}+\mathrm{y\text{'}\text{'}}\ne 0\right],\left[\mathrm{y\text{'}}=-\frac{y^4}{4}\right]\mathbf{where}\left[y\ne 0\right],\left[\mathrm{y\text{'}}=0\right]\mathbf{where}\left[y\ne 0\right],\left[y=0\right]\mathbf{where}\left[\right]$

sys[1] := {diff(y(t),t)=-x(t)^2,diff(x(t),t)=y(t)};

${\mathrm{sys}}_1≔\left\{\mathrm{x\text{'}}=y\,\mathrm{y\text{'}}=-{x\left(t\right)}^2\right\}$

infolevel[:-casesplit] := 2;

${\mathrm{infolevel}}_{\mathrm{casesplit}}≔2$

casesplit(sys[1]);

-> Solving ordering to be used: [y, x]

Ranking for 1st round is: [[y, x]]
Ranking for 2nd round is: [y, x]

1st round is performed, original system is split into 1 cases

2nd round is performed; processing answer ...

$\left[y=\mathrm{x\text{'}}\,\mathrm{x\text{'}\text{'}}=-{x\left(t\right)}^2\right]\mathbf{where}\left[\right]$

[casesplit(sys[1],[x, y])];

Ranking for 1st round is: [[x, y]]
Ranking for 2nd round is: [x, y]

1st round is performed, original system is split into 1 cases

2nd round is performed, original system is now split into 2 cases. Processing answer ...

$\left[\left[x\left(t\right)=-\frac{\mathrm{y\text{'}\text{'}}}{2y}\,\mathrm{y\text{'}\text{'}\text{'}}=\frac{-2y^3+\mathrm{y\text{'}\text{'}}\mathrm{y\text{'}}}{y}\,{\mathrm{y\text{'}\text{'}}}^2=-4\mathrm{y\text{'}}{y}^2\right]\mathbf{where}\left[y\ne 0\right]\,\left[x\left(t\right)=0\,y=0\right]\mathbf{where}\left[\right]\right]$

sys[2] := {diff(g(t),t) = -f(t)^(1/2), diff(j(t),t) = -h(t),
diff(f(t),t) = h(t)+g(t), diff(diff(h(t),t),t) = j(t)+g(t)^(1/2)};

${\mathrm{sys}}_2≔\left\{\mathrm{h\text{'}\text{'}}=j+\sqrt{g}\,\mathrm{f\; \text{'}}=h+g\,\mathrm{g\text{'}}=-\sqrt{f}\,\mathrm{j\text{'}}=-h\right\}$

casesplit(sys[2]);

-> Solving ordering to be used: [j, h]

Ranking for 1st round is: [[j, h, f, _F1, g, _F2]]
Ranking for 2nd round is: [j, h, f, _F1, g, _F2]

1st round is performed, original system is split into 2 cases

2nd round is performed; processing answer ...

$\left[j=2\mathrm{g\text{'}\text{'}\text{'}\text{'}}\mathrm{g\text{'}}+6\mathrm{g\text{'}\text{'}\text{'}}\mathrm{g\text{'}\text{'}}-\mathrm{g\text{'}\text{'}}-\sqrt{g}\,h=2\mathrm{g\text{'}}\mathrm{g\text{'}\text{'}}-g\,\sqrt{f}=-\mathrm{g\text{'}}\,\mathrm{g\text{'}\text{'}\text{'}\text{'}\text{'}}=-\frac{16\mathrm{g\text{'}\text{'}\text{'}\text{'}}\mathrm{g\text{'}\text{'}}+12{\mathrm{g\text{'}\text{'}\text{'}}}^2+4\mathrm{g\text{'}}\mathrm{g\text{'}\text{'}}-\frac{\mathrm{g\text{'}}}{\sqrt{g}}-2\mathrm{g\text{'}\text{'}\text{'}}-2g}{4\mathrm{g\text{'}}}\right]\mathbf{where}\left[\mathrm{g\text{'}}\ne 0\right],\left[j=0\,h=0\,f=0\,g=0\right]\mathbf{where}\left[\right]$

infolevel[:-casesplit] := 1;

${\mathrm{infolevel}}_{\mathrm{casesplit}}≔1$

f(x) = exp(BesselJ(nu, sqrt(x)));

$f\left(x\right)={\ⅇ}^{\mathrm{BesselJ}\left(\mathrm{\nu }\,\sqrt{x}\right)}$

dpolyform((12), no_Fn);

odetest((12), (13));

$\left[0\right]$

F(y-I*x) + G(y+I*x) = H(x);

$F\left(y-\mathrm{I}x\right)+G\left(y+\mathrm{I}x\right)=H\left(x\right)$

casesplit((15));

$\left[H\left(x\right)=F\left(y-\mathrm{I}x\right)+G\left(y+\mathrm{I}x\right)\right]\mathbf{where}\left[\right]$

casesplit((15), [F, G]);

$\left[F\left(y-\mathrm{I}x\right)=-G\left(y+\mathrm{I}x\right)+H\left(x\right)\,\mathrm{D}\left(G\right)\left(y+\mathrm{I}x\right)=-\frac{\mathrm{I}}{2}{H}_x\right]\mathbf{where}\left[\right]$

F(x, t) = H(t*x+G(1/t));

$F\left(x\,t\right)=H\left(tx+G\left(\frac{1}{t}\right)\right)$

dpolyform((18), no_Fn);

$\left[F_{t,x}=\frac{F_t{F}_{x,x}}{F_x}+\frac{F_x}{t}\right]\mathbf{where}\left[F_x\ne 0\right]$

pdetest((18), (19));

$\left[0\right]$

pde := diff(u(x, t), x, t) = D(D(D(f)))(x - t) ;

$\mathrm{pde}≔u_{t,x}={\mathrm{D}}^{\left(3\right)}\left(f\right)\left(x-t\right)$

dpolyform(pde, no_Fn);

$\left[u_{t,x,x}=-u_{t,t,x}\right]\mathbf{where}\left[\right]$

pdsolve((22));

$\left\{u\left(x\,t\right)=\mathrm{f\_\_1}\left(t\right)+\mathrm{f\_\_2}\left(x\right)+\mathrm{f\_\_3}\left(t-x\right)\right\}$

pde[1] := diff(f(x,y,z),x,y)*diff(f(x,y,z),y)=h(x)*g(y,z);

${\mathrm{pde}}_1≔f_{x,y}{f}_y=h\left(x\right)g\left(y\,z\right)$

[casesplit(pde[1], f)];  # [f] or [f(x,y,z)]

$\left[\left[f_{x,y}=\frac{h\left(x\right)g\left(y\,z\right)}{f_y}\right]\mathbf{where}\left[f_y\ne 0\right]\,\left[f_y=0\,h\left(x\right)=0\right]\mathbf{where}\left[g\left(y\,z\right)\ne 0\right]\,\left[f_y=0\,g\left(y\,z\right)=0\right]\mathbf{where}\left[\right]\right]$

pde[2] := [diff(u(x,y),x) + v(x,y) = y,
diff(u(x,y),y)+v(x,y)=0,
diff(v(x,y),x) = diff(v(x,y),y)];

${\mathrm{pde}}_2≔\left[u_x+v\left(x\,y\right)=y\,u_y+v\left(x\,y\right)=0\,v_x=v_y\right]$

casesplit(pde[2]);

Warning: System is inconsistent

declare(y(x), prime=x);

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

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

ode[3] := diff(y(x),x,x)=y(x)*diff(y(x),x)*x;

${\mathrm{ode}}_3≔\mathrm{y\text{'}\text{'}}=y\mathrm{y\text{'}}x$

odepde(ode[3], _mu = mu(x,y(x)));

${\mathrm{\_y1}}^2{\mathrm{\mu }}_{y,y}+2{\mathrm{\mu }}_{x,y}\mathrm{\_y1}+y\mathrm{\mu }\left(x\,y\right)+2\mathrm{\_y2}{\mathrm{\mu }}_y+{\mathrm{\mu }}_{x,x}+{\mathrm{\mu }}_{x}yx$

sys[3] := [ gensys(ode[3], _mu = mu(x,y)) ];

${\mathrm{sys}}_3≔\left[{\mathrm{\mu }}_y\,{\mathrm{\mu }}_{y,y}\,{\mathrm{\mu }}_{x,y}\,y\mathrm{\mu }\left(x\,y\right)+{\mathrm{\mu }}_{x,x}+{\mathrm{\mu }}_{x}yx\right]$

casesplit(sys[3]);

$\left[\mathrm{\mu }\left(x\,y\right)=0\right]\mathbf{where}\left[\right]$

sys[4] := [ gensys(ode[3], _mu = mu(x,diff(y(x),x))) ];

${\mathrm{sys}}_4≔\left[{\mathrm{\mu }}_{\mathrm{\_y1},\mathrm{\_y1}}x\mathrm{\_y1}+2{\mathrm{\mu }}_{\mathrm{\_y1}}x\,{\mathrm{\mu }}_{\mathrm{\_y1},x}\,{\mathrm{\mu }}_{\mathrm{\_y1},x}x\mathrm{\_y1}+{\mathrm{\mu }}_{\mathrm{\_y1}}\mathrm{\_y1}+{\mathrm{\mu }}_{x}x+\mathrm{\mu }\left(x\,\mathrm{\_y1}\right)\,{\mathrm{\mu }}_{\mathrm{\_y1}}{\mathrm{\_y1}}^{2}x+{\mathrm{\mu }}_{x,x}\right]$

casesplit(sys[4]);

$\left[\mathrm{\mu }\left(x\,\mathrm{\_y1}\right)=0\right]\mathbf{where}\left[\right]$

sys[5] := [ gensys(ode[3], _mu = mu(x,diff(y(x),x))) ];

${\mathrm{sys}}_5≔\left[{\mathrm{\mu }}_{\mathrm{\_y1},\mathrm{\_y1}}x\mathrm{\_y1}+2{\mathrm{\mu }}_{\mathrm{\_y1}}x\,{\mathrm{\mu }}_{\mathrm{\_y1},x}\,{\mathrm{\mu }}_{\mathrm{\_y1},x}x\mathrm{\_y1}+{\mathrm{\mu }}_{\mathrm{\_y1}}\mathrm{\_y1}+{\mathrm{\mu }}_{x}x+\mathrm{\mu }\left(x\,\mathrm{\_y1}\right)\,{\mathrm{\mu }}_{\mathrm{\_y1}}{\mathrm{\_y1}}^{2}x+{\mathrm{\mu }}_{x,x}\right]$

casesplit(sys[5]);

$\left[\mathrm{\mu }\left(x\,\mathrm{\_y1}\right)=0\right]\mathbf{where}\left[\right]$

ode[4] := diff(y(x),x,x) - 3*y(x)*diff(y(x),x) - 3*a*y(x)^2 - 4*a^2*y(x) - b = 0;

${\mathrm{ode}}_4≔\mathrm{y\text{'}\text{'}}-3y\mathrm{y\text{'}}-3a{y}^2-4a^{2}y-b=0$

sys[6] := [ gensys(ode[4], [xi(x,y), eta(x,y)]) ];

${\mathrm{sys}}_6≔\left[{\mathrm{\xi }}_{y,y}\,-6{\mathrm{\xi }}_{y}y+{\mathrm{\eta }}_{y,y}-2{\mathrm{\xi }}_{x,y}\,-12{\mathrm{\xi }}_y{a}^{2}y-9{\mathrm{\xi }}_{y}a{y}^2-3{\mathrm{\xi }}_{y}b-3{\mathrm{\xi }}_{x}y-3\mathrm{\eta }\left(x\,y\right)+2{\mathrm{\eta }}_{x,y}-{\mathrm{\xi }}_{x,x}\,-8{\mathrm{\xi }}_x{a}^{2}y-6{\mathrm{\xi }}_{x}a{y}^2+4{\mathrm{\eta }}_y{a}^{2}y+3{\mathrm{\eta }}_{y}a{y}^2-4\mathrm{\eta }\left(x\,y\right)a^2-6\mathrm{\eta }\left(x\,y\right)ay-2{\mathrm{\xi }}_{x}b+{\mathrm{\eta }}_{y}b-3{\mathrm{\eta }}_{x}y+{\mathrm{\eta }}_{x,x}\right]$

casesplit(sys[6]);

$\left[\mathrm{\eta }\left(x\,y\right)=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\xi }}_y=0\right]\mathbf{where}\left[\right]$

ode[5] := diff(y(x),x,x) - diff(y(x),x)^2/y(x) - g(x)*y(x)*diff(y(x),x) - diff(g(x),x)*y(x)^2 = 0;

${\mathrm{ode}}_5≔\mathrm{y\text{'}\text{'}}-\frac{{\mathrm{y\text{'}}}^2}{y}-g\left(x\right)y\mathrm{y\text{'}}-\mathrm{g\text{'}}{y}^2=0$

sys[7] := gensys(ode[5], [xi(x,y), eta(x,y)], y(x));

${\mathrm{sys}}_7≔-{\mathrm{\xi }}_{y,y}{y}^2-{\mathrm{\xi }}_{y}y,-2g\left(x\right){\mathrm{\xi }}_y{y}^3+{\mathrm{\eta }}_{y,y}{y}^2-2{\mathrm{\xi }}_{x,y}{y}^2-{\mathrm{\eta }}_{y}y+\mathrm{\eta }\left(x\,y\right),-3\mathrm{g\text{'}}{\mathrm{\xi }}_y{y}^4-g\left(x\right){\mathrm{\xi }}_x{y}^3-\mathrm{g\text{'}}\mathrm{\xi }\left(x\,y\right)y^3-g\left(x\right)\mathrm{\eta }\left(x\,y\right)y^2+2{\mathrm{\eta }}_{x,y}{y}^2-{\mathrm{\xi }}_{x,x}{y}^2-2{\mathrm{\eta }}_{x}y,-2\mathrm{g\text{'}}{\mathrm{\xi }}_x{y}^4+\mathrm{g\text{'}}{\mathrm{\eta }}_y{y}^4-\mathrm{\xi }\left(x\,y\right)\mathrm{g\text{'}\text{'}}{y}^4-g\left(x\right){\mathrm{\eta }}_x{y}^3-2\mathrm{g\text{'}}\mathrm{\eta }\left(x\,y\right)y^3+{\mathrm{\eta }}_{x,x}{y}^2$

ineqs[7] := xi(x,y) <> 0, eta(x,y) <> 0;

${\mathrm{ineqs}}_7≔\mathrm{\xi }\left(x\&comma;y\right)\ne 0,\mathrm{\eta }\left(x\&comma;y\right)\ne 0$

[casesplit([sys[7], ineqs[7]], [ [xi,eta], g ])];

$\left[\left[\mathrm{\xi }\left(x\&comma;y\right)=\frac{2{g\left(x\right)}^2\mathrm{\eta }\left(x\&comma;y\right)\mathrm{g\text{'}\text{'}}-2g\left(x\right)\mathrm{\eta }\left(x\&comma;y\right){\mathrm{g\text{'}}}^2}{{g\left(x\right)}^2\mathrm{g\text{'}\text{'}\text{'}}y-5g\left(x\right)\mathrm{g\text{'}}\mathrm{g\text{'}\text{'}}y+4{\mathrm{g\text{'}}}^{3}y}\&comma;{\mathrm{\eta }}_x=\frac{\mathrm{\eta }\left(x\&comma;y\right)g\left(x\right)\mathrm{g\text{'}\text{'}\text{'}}\mathrm{g\text{'}}-2\mathrm{\eta }\left(x\&comma;y\right)g\left(x\right){\mathrm{g\text{'}\text{'}}}^2+\mathrm{\eta }\left(x\&comma;y\right){\mathrm{g\text{'}}}^2\mathrm{g\text{'}\text{'}}}{{g\left(x\right)}^2\mathrm{g\text{'}\text{'}\text{'}}-5g\left(x\right)\mathrm{g\text{'}}\mathrm{g\text{'}\text{'}}+4{\mathrm{g\text{'}}}^3}\&comma;{\mathrm{\eta }}_y=\frac{\mathrm{\eta }\left(x\&comma;y\right)}{y}\&comma;\mathrm{g\text{'}\text{'}\text{'}\text{'}}=\frac{3{g\left(x\right)}^2{\mathrm{g\text{'}\text{'}\text{'}}}^2-10g\left(x\right)\mathrm{g\text{'}}\mathrm{g\text{'}\text{'}}\mathrm{g\text{'}\text{'}\text{'}}+6g\left(x\right){\mathrm{g\text{'}\text{'}}}^3+4{\mathrm{g\text{'}}}^3\mathrm{g\text{'}\text{'}\text{'}}-3{\mathrm{g\text{'}}}^2{\mathrm{g\text{'}\text{'}}}^2}{2{g\left(x\right)}^2\mathrm{g\text{'}\text{'}}-2g\left(x\right){\mathrm{g\text{'}}}^2}\right]\mathbf{where}\left[\mathrm{\eta }\left(x\&comma;y\right)\ne 0\&comma;{g\left(x\right)}^2\mathrm{g\text{'}\text{'}\text{'}}-5g\left(x\right)\mathrm{g\text{'}}\mathrm{g\text{'}\text{'}}+4{\mathrm{g\text{'}}}^3\ne 0\right]\&comma;\left[{\mathrm{\xi }}_x=-\frac{\mathrm{g\text{'}}\mathrm{\xi }\left(x\&comma;y\right)y+g\left(x\right)\mathrm{\eta }\left(x\&comma;y\right)}{g\left(x\right)y}\&comma;{\mathrm{\xi }}_y=0\&comma;{\mathrm{\eta }}_x=\frac{{\mathrm{g\text{'}}}^2\mathrm{\xi }\left(x\&comma;y\right)y+g\left(x\right)\mathrm{\eta }\left(x\&comma;y\right)\mathrm{g\text{'}}}{{g\left(x\right)}^2}\&comma;{\mathrm{\eta }}_y=\frac{\mathrm{\eta }\left(x\&comma;y\right)}{y}\&comma;\mathrm{g\text{'}\text{'}}=\frac{{\mathrm{g\text{'}}}^2}{g\left(x\right)}\right]\mathbf{where}\left[\mathrm{\eta }\left(x\&comma;y\right)\ne 0\&comma;g\left(x\right)\ne 0\&comma;\mathrm{\xi }\left(x\&comma;y\right)\ne 0\right]\&comma;\left[{\mathrm{\xi }}_{x,x}=\frac{2{\mathrm{\eta }}_{x,y}y-2{\mathrm{\eta }}_x}{y}\&comma;{\mathrm{\xi }}_{x,y}=-\frac{-{\mathrm{\eta }}_{y,y}{y}^2+{\mathrm{\eta }}_{y}y-\mathrm{\eta }\left(x\&comma;y\right)}{2y^2}\&comma;{\mathrm{\xi }}_{y,y}=-\frac{{\mathrm{\xi }}_y}{y}\&comma;{\mathrm{\eta }}_{x,y,y}=\frac{{\mathrm{\eta }}_{x,y}y-{\mathrm{\eta }}_x}{y^2}\&comma;{\mathrm{\eta }}_{y,y,y}=\frac{-{\mathrm{\eta }}_{y}y+\mathrm{\eta }\left(x\&comma;y\right)}{y^3}\&comma;{\mathrm{\eta }}_{x,x}=0\&comma;g\left(x\right)=0\right]\mathbf{where}\left[\mathrm{\eta }\left(x\&comma;y\right)\ne 0\&comma;\mathrm{\xi }\left(x\&comma;y\right)\ne 0\right]\right]$

sys[8] := [-diff(mu(x,y),y)-A(y)*mu(x,y)=0,
3*mu(x,y)*F(x)*y^2 - diff(mu(x,y),x) + diff(mu(x,y),y)*H(x) + diff(mu(x,y),y)*F(x)*y^3-B(x)*mu(x,y) = 0,
mu(x,y)*diff(H(x),x) + diff(mu(x,y),x)*F(x)*y^3 + diff(mu(x,y),x)*H(x) + mu(x,y)*diff(F(x),x)*y^3 = 0];

${\mathrm{sys}}_8≔\left[-{\mathrm{\mu }}_y-A\left(y\right)\mathrm{\mu }\left(x\&comma;y\right)=0\&comma;3\mathrm{\mu }\left(x\&comma;y\right)F\left(x\right)y^2-{\mathrm{\mu }}_x+{\mathrm{\mu }}_{y}H\left(x\right)+{\mathrm{\mu }}_{y}F\left(x\right)y^3-B\left(x\right)\mathrm{\mu }\left(x\&comma;y\right)=0\&comma;\mathrm{\mu }\left(x\&comma;y\right)\mathrm{H\text{'}}+{\mathrm{\mu }}_{x}F\left(x\right)y^3+{\mathrm{\mu }}_{x}H\left(x\right)+\mathrm{\mu }\left(x\&comma;y\right)\mathrm{F\text{'}}{y}^3=0\right]$

casesplit(sys[8], rankingused = 'R', ivarsused = 'ii', singsol = false);

$\left[B\left(x\right)=\frac{3{\mathrm{\mu }\left(x\&comma;y\right)}^{2}F\left(x\right){\mathrm{\mu }}_{y,y}{y}^2-6\mathrm{\mu }\left(x\&comma;y\right)F\left(x\right){{\mathrm{\mu }}_y}^2{y}^2-6{\mathrm{\mu }\left(x\&comma;y\right)}^{2}F\left(x\right){\mathrm{\mu }}_{y}y-\mathrm{\mu }\left(x\&comma;y\right){\mathrm{\mu }}_{y,y}{\mathrm{\mu }}_x+{{\mathrm{\mu }}_y}^2{\mathrm{\mu }}_x}{{\mathrm{\mu }\left(x\&comma;y\right)}^2{\mathrm{\mu }}_{y,y}-\mathrm{\mu }\left(x\&comma;y\right){{\mathrm{\mu }}_y}^2}\&comma;H\left(x\right)=\frac{-\mathrm{\mu }\left(x\&comma;y\right)F\left(x\right){\mathrm{\mu }}_{y,y}{y}^3+F\left(x\right){{\mathrm{\mu }}_y}^2{y}^3-3{\mathrm{\mu }}_y\mathrm{\mu }\left(x\&comma;y\right)F\left(x\right)y^2-6{\mathrm{\mu }\left(x\&comma;y\right)}^{2}F\left(x\right)y}{{\mathrm{\mu }}_{y,y}\mathrm{\mu }\left(x\&comma;y\right)-{{\mathrm{\mu }}_y}^2}\&comma;\mathrm{F\text{'}}=-\frac{F\left(x\right){\mathrm{\mu }}_x}{\mathrm{\mu }\left(x\&comma;y\right)}\&comma;A\left(y\right)=-\frac{{\mathrm{\mu }}_y}{\mathrm{\mu }\left(x\&comma;y\right)}\&comma;{\mathrm{\mu }}_{y,y,y}=\frac{2\mathrm{\mu }\left(x\&comma;y\right){{\mathrm{\mu }}_{y,y}}^2{y}^2-{\mathrm{\mu }}_{y,y}{{\mathrm{\mu }}_y}^2{y}^2+8\mathrm{\mu }\left(x\&comma;y\right){\mathrm{\mu }}_{y,y}{\mathrm{\mu }}_{y}y-6{{\mathrm{\mu }}_y}^{3}y+2{\mathrm{\mu }\left(x\&comma;y\right)}^2{\mathrm{\mu }}_{y,y}-2\mathrm{\mu }\left(x\&comma;y\right){{\mathrm{\mu }}_y}^2}{{\mathrm{\mu }}_y\mathrm{\mu }\left(x\&comma;y\right)y^2+2{\mathrm{\mu }\left(x\&comma;y\right)}^{2}y}\&comma;{\mathrm{\mu }}_{x,y}=\frac{{\mathrm{\mu }}_y{\mathrm{\mu }}_x}{\mathrm{\mu }\left(x\&comma;y\right)}\right]\mathbf{where}\left[F\left(x\right)\ne 0\&comma;{\mathrm{\mu }}_{y}y+2\mathrm{\mu }\left(x\&comma;y\right)\ne 0\&comma;-{\mathrm{\mu }}_{y,y}\mathrm{\mu }\left(x\&comma;y\right)+{{\mathrm{\mu }}_y}^2\ne 0\right]$

R, ii;

$\left[B\&comma;H\&comma;F\&comma;A\&comma;\mathrm{\mu }\right],\left[\left[y\&comma;x\right]\right]$

[ [mu,A, B], [F, H] ];

$\left[\left[\mathrm{\mu }\&comma;A\&comma;B\right]\&comma;\left[F\&comma;H\right]\right]$

DR := diffalg[differential_ring](notation = diff, derivations = [x,y], ranking = [[mu, A, B], [F, H]]);

$\mathrm{DR}≔\mathrm{PDE\_ring}$

diffalg[print_ranking](DR);

In lists, leftmost elements are greater than rightmost ones.
The derivatives of [mu, A, B] are ordered by grlexA:
_U [tau] > _V [phi] when
    |tau| > |phi| or
    |tau| = |phi| and _U > _V w.r.t. the list of indeterminates or
    |tau| = |phi| and _U = _V and tau > phi w.r.t. [x, y]
Any derivative of [mu, A, B] is greater than any derivative of [F, H]
The derivatives of [F, H] are ordered by grlexA.

with(Physics):

Setup(anticommutativepre = {theta, Q});

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\&comma;\mathrm{\theta }\right\}\right]$

PDEtools:-declare(Q(x, y, theta[1], theta[2]));

$Q\left(x\&comma;y\&comma;{\mathrm{\theta }}_1\&comma;{\mathrm{\theta }}_2\right)\mathrm{will\; now\; be\; displayed\; as}Q$

q := PDEtools:-diff_table(Q(x, y, theta[1], theta[2])):

pde[1] := q[x, y, theta[1]] + q[x, y, theta[2]] - q[y, theta[1], theta[2]] = 0;

${\mathrm{pde}}_1≔Q_{x,y,{\mathrm{\theta }}_1}+Q_{x,y,{\mathrm{\theta }}_2}-Q_{y,{\mathrm{\theta }}_1,{\mathrm{\theta }}_2}=0$