The best software is Matlab but is not free, so you can use Derive or Octave, those are mathematical assistants. I know Octave is free, you can looking for is Derive is free too or not.

You can also consider using Maxima and Reduce, both of which are free:

http://sourceforge.net/projects/reduce/

http://sourceforge.net/projects/maxima/

(you can also try Maxima online for free at maxima-online.org)

For some simpler tasks you can also try to use Wolfram Alpha online for free:

http://wolframalpha.com

Speaking of commercially available software, you can consider Maple or Mathematica.

It looks like R has packages for solving differential equations: http://journal.r-project.org/archive/2010-2/RJournal_2010-2_Soetaert~et~al.pdf. I've never tried these, but may be worth a try

Depending on your programming skills I would recommend the following two software packages:

- COPASI (http://www.copasi.org):

easy to use, graphical user interface, stand-alone software package.

- Data2Dynamics (https://bitbucket.org/d2d-development/d2d-software/):

Advanced methods are implemented, no graphical interface, computationally more efficient for parameter estimation and identifiability analysis, requires MATLAB installation and some MATLAB skills.

Thank you, everyone, for your quick and useful responses!

Use comsol

XPP? Or pretty much any fast stand-alone ODE solver combined with a scripting language to do the optimization (R, scipy, matlab. SciLab is actually free)

you first must decide on what you mean by ''fit'': your ode will have parameters,

maybe the initial values are also required, and you have measurement data from this

proces. You now must decide on the error measure : least squares? or max abs deviation? or .. then you end up with a (constrained) optimization problem

Constraints will typically concern range of parameters, exact fit of special values

etc.

for example sum_over the measurement grid { solution of ode - measurement }^2

In this case easyfit from Klaus Schittkowski would be a good tool (with nice interface

for windows), no much programming from your side.

In principle you can use any available optimizer and your error measure plus the ode solver will play the part of the user function input. For a list of optimization codes

(nonlinear constrained , your case) look here

http://plato.asu.edu/guide.html

if your system is small and the data not too much then you can try this concept here:

http://numawww.mathematik.tu-darmstadt.de

in the section least squares :DGLFIT

Thanks a lot, Alexander and Peter!

SAGE (python based) offers you ODE solver and a lot of analysis tools.

http://www.sagemath.org/

JSim is designed for this mission. Free. Linux, Mac and Windows. With it one can go from experinenbt design through all the steps of data analysis, verification and validation testing to uncertainty quantification, the whole VVUQ process.

Butterworth E, Raymond GM, Jardine B, Neal ML, and Bassingthwaighte JB. JSim, an open-source modeling system for data analysis. F1000Research 2: 288, 2013.

JSim is a simulation system for developing models, designing experiments, and evaluating hypotheses on physiological and pharmacological systems through the testing of model solutions against data. It is designed for interactive, iterative manipulation of the model code, handling of multiple data sets and parameter sets, and for making comparisons among different models running simultaneously or separately. Interactive use is supported by a large collection of graphical user interfaces for model writing, compilation diagnostics, defining input functions, model runs, selection of algorithms solving ordinary and partial differential equations, run-time multidimensional graphics, parameter optimization (8 methods), sensitivity analysis, and Monte Carlo simulation for defining confidence ranges. JSim uses Mathematical Modeling Language (MML) a declarative syntax specifying algebraic and differential equations. Imperative constructs written in other languages (MATLAB, FORTRAN, C++, etc.) are accessed through procedure calls. MML syntax is simple, basically defining the parameters and variables, then writing the equations in a straightforward, easily read and understood mathematical form. This makes JSim good for teaching modeling as well as for model analysis for research. For high throughput applications, JSim can be run as a batch job. JSim can automatically translate models from the repositories for Systems Biology Markup Language (SBML) and CellML models. Stochastic modeling is supported. MML supports assigning physical units to constants and variables and automates checking dimensional balance as the first step in verification testing. Automatic unit scaling follows, e.g. seconds to minutes, if needed. The JSim Project File sets a standard for reproducible modeling analysis: it includes in one file everything for analyzing a set of experiments: the data, the models, the data fitting, and evaluation of parameter confidence ranges. JSim is open source; it and about 400 human readable open source physiological /biophysical models are available at http://www.physiome.org/jsim/. Experimentalists wishing to analyze their own data with the models from this repository can simply import their data and evaluate them via the model, optimizing to find best fitting parameter values and their confidence ranges and sensitivities.

There are some tools in Matlab, e.g. SBtoolbox2 and Potterswheel.

"LS Optimizer Software" solves inverse problem once you create a solver (file .exe) for the direct problem: function, ODE, PDE etc. 

Minitab or Curve Expert Pro can help you to do this job.