![]() The union of the non-singular solutions of each of these ODEs and algebraic equations is equal to the set of general and singular solutions of the original ODE system. When the ODE system, rational in the unknowns and their derivatives, contains non-polynomial coefficients, the triangularization-also called sequential uncoupling-is performed by rewriting the system in polynomial form by using a differential extension approach (see PDEtools ).ĭue to the intrinsic nature of the differential algebra elimination process, this first step always produces (possibly many) ODEs or algebraic equations, such that: ![]() The system is first triangularized by using differential algebra techniques for polynomial systems. The solving process consists of two steps.ġ. ODE systems rational in the unknowns: The solving process However, step 1) is systematic in that, whenever the system is consistent, the uncoupling of systems rational in the unknowns and their derivatives is feasible (up to the amount of memory available on your computer). Depending on the complexity of the system, step 2) may not be fully successful. Given an ODE system, possibly including algebraic constraints and inequations, dsolve returns an exact solution when: 1) it succeeds in uncoupling the system and 2) dsolve 's routines for solving a single ODE succeed in solving each of the ODEs arising in the uncoupling process. Rif - request that the differential elimination step for nonlinear ODE systems should also be done using the DEtools package Parameters=P - P is a list or set of names or functions which are solving variables with priority lower than any other variable Mindim=N - avoid the computation of solutions when the dimension of the solution space is less than N Singsol=false - avoid the computation of the singular solutions when the system is nonlinear UseInt - request the use of Inert integrals instead of performing the integrations involved in the solution (optional) argument(s) that can be given in any order and are described in Optionsįuncs - set or list with indeterminate functions or just their names - can also be a rankingĮxplicit - request the composition of the sets appearing in the solution in the case of nonlinear systems of ODEs System of ODEs it can contain inequations Find exact solutions for systems of ordinary differential equations (ODEs)
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