The first argument sys can be one DE, or a set or a list of DEs, possibly including inequations and non-differential equations (such as algebraic constraints). Equations are entered by using , or expr1. In this case it is assumed that expr1 = 0. Inequations are entered by using .

The output of casesplit is a sequence of expressions representing the different cases (possibly only one) and corresponding equations and inequations (these are the ones returned by the corresponding Equations and Inequations commands of DifferentialAlgebra). Each case is returned as a function of two operands and displayed as . Both operands are lists. This printing is obtained, however, by using a print/foo mechanism, and the true structure of each returned case is a function `casesplit/ans`( [ eq1, eq2, ...], [ ineq1 <> 0, ineq2 <> 0, ...] ).

The actual splitting of a system obtained when calling casesplit is basically computed through various calls to different commands of the DifferentialAlgebra package, or optionally the DEtools[Rif] package. These are the two Maple differential elimination packages.

Apart from simplifying the steps required by a direct use of DifferentialAlgebra, casesplit radically extends the domain of computation of DifferentialAlgebra or DEtools[Rif]. Concretely, casesplit can split a system into regular cases even when this system contains non-polynomial objects, such as fractional or abstract powers, most Maple known functions (for example, exponentials, logarithms, trigonometric, and special functions) and also any arbitrary composition of these non-polynomial objects, for example, .

This extension of the domain of computation is performed by rewriting all non-polynomial objects by using auxiliary functions of the form  (n being an integer) and adding to the input system the polynomial (algebraic and differential) equations these  functions satisfy (see dpolyform). For example,  is replaced by , and the auxiliary ODE  is added to the system. These auxiliary functions, the equations they satisfy, and their consequences are removed before returning the output.

All arguments after the first one are optional and can be included in any order.

When the solving variables are not specified, casesplit takes all unknown functions found in the DE system as solving variables, and uses its own heuristic methods to determine what could be the best uncoupling ordering for the given system. However, it is possible to specify a ranking, which indicates the solving variables and their order. When the ranking to be specified involves only one variable, it can be given as this variable, or its name, for example,  for . In all cases the ranking can be specified as a list or a set of solving variables, functions can be indicated by only their names, and this set or list can include 'symbol' solving variables (equivalent to functions with all derivatives equal to zero).

When the ranking is given as a set or list, each of its elements can, in turn, also be a set or list of names or functions. For example, , , and  are valid rankings. In such cases, the solving ordering for the elements inside each set  is decided by casesplit. Only one level of nested list or sets is allowed.

Using standard terminology, a ranking as in  - also called lexicographic or elimination ranking - leads to a splitting using a lexicographic ordering. When a ranking is given as a nested list, for example , the splitting is performed by using a total degree ordering. These rankings are also called total degree or orderly rankings. Total degree rankings, however, generally do not result in the uncoupling of the given system. They simplify the system by using integrability conditions, and so check for inconsistencies as well as lead to a version of the system with the lowest differential order. This usually leads to faster computations than when using lexicographic ordering. To understand the meaning associated with a given ranking, use the checkrank or DifferentialAlgebra[Tools][Display] commands. Input rankings in casesplit have exactly the same meaning they have in calls to commands of the DifferentialAlgebra package.

When the ranking is not specified, you can see the ranking used by setting infolevel[casesplit] := 2. Setting this variable to  causes the display of the input sent to RosenfeldGroebner or DEtools[rifsimp], as well.

The independent variables of a given system are determined by default as the variables on which the unknown functions of the system depend. However, there may be cases in which other variables, not present in the unknown functions of the system, are also independent variables of the problem. These extra independent variables can be indicated using the ivars = {...} optional argument. In such a case, these variables are added to those determined as the independent variables of the problem. This is useful, for example, when the unknowns of the system depend - say - on x, y, but the equations involve - say - z, and you are not interested in solutions depending on z.

By default, the solving ordering for the independent variables of the system is automatically determined by casesplit, such that lower order differentiation variables are ranked lower. Alternatively, this ordering can be enforced to be anyone using the optional argument ivars = [...], specifying its right-hand side as a list with the desired ordering, leading in most of the cases to slight differences in the output (see DifferentialAlgebra[Tools][Display]).

By default, the ranking used with respect to the independent variables is of orderly type (see paragraphs above). To specify an elimination lexicographical ranking add the keyword anywhere in the calling sequence. Combining this option with the ivars = [...] option (right-hand side as a list) is useful to generate ODEs from PDE systems when that is possible. The ODEs appearing in the output are the consequence of using an elimination ranking for the independent variables. When ivars = [...] is not given, but lex is given, the elimination raking is set with the ordering for the independent variables chosen by casesplit; or when lex(f) is given, where f is an unknown of the system, the ordering used for ranking the independent variables will be that of the function f.

By default, all unknown functions found in the system and not indicated in the ranking are considered solving variables "with less priority", and are automatically inserted at the end of the ranking. Note however that symbol variables found in the system require an explicit declaration in the ranking to be considered as "solving variables". "Solving variables" with less priority can also be indicated using the optional argument parameters={...} where the right-hand side can also be a list.

In the case of DE systems, the default strategy of casesplit is to solve the system in two rounds. In the first round, a simpler ranking is used to express the more differentiated solving variables with respect to the other ones. Typically this ranking is built as , and relies on a faster "total degree" process. In the second round, each of the systems (equations and inequations) obtained in the first round are processed separately, but this time with the ranking given by the user (or set by casesplit if no ranking is given). When the optional argument 'fast' is given, casesplit processes the system in only one round. In such a case, if the ranking is given it is used. Otherwise, the ranking is set as  (that is, total degree ordering) and usually does not lead to an uncoupling of the system.

By default, casesplit splits a nonlinear problem into all the related (perhaps only one) regular cases. This set of regular cases can also be understood as consisting of the main case (general case) plus all the possible singular cases. To avoid spending time in computing these singular cases, one can use the singsol=false optional argument, leading to the computation of only the general case.

By using the optional argument arbitrary=A, where A is a list or set of names or functions, one can force casesplit to avoid considering these functions as solving variables and so avoid splitting into cases with respect to them. This option is quite useful for computations in which the value of some unknown functions must remain arbitrary. It also typically reduces the computation time significantly.

By default, casesplit splits a problem into cases regardless of the time required to compute each case. There are problems, however, for which this default is not practical since some of the cases would take too much time to compute. For these cases, use the ctl=N optional argument to impose a timelimit of N seconds per case, at the cost of missing the output of cases requiring more than N seconds to be computed.

The optional argument mindim=N is useful when we know, in advance, that we are not interested in cases in which the solution space has dimension less than N. This option saves computing time.

If rankingused = .. is passed as argument, where the right-hand side is assignable (e.g., a name), the ranking used for the unknowns will be assigned to that right-hand side.

If ivarsused = .. is passed as argument, where the right-hand side is assignable (e.g., a name), the independent variables of the system, with the ordering they were ranked, will be assigned to that right-hand side.

By default, casesplit makes use of the Rif subpackage of DEtools. In most cases, however, it is possible to perform the same computations by using the DifferentialAlgebra package instead of Rif, by giving the extra argument diffalg. Any of the following optional arguments imply on computations which casesplit can only run using Rif: singsol=false, arbitrary, ctl=N.