Karline Soetaert was the driving force behind five packages that make R quite a versatile tool for solving differential equations. Thanks to this work, it is now possible to solve initial value problems of ordinary differential equations, delay differential equations, differential algebraic equations, partial differential equations, and boundary value problems in R.
The main package is deSolve, developed with Thomas Petzoldt from Germany and Woody Setzer from the US; two other packages, bvpSolve and deTestSet were developed with Jeff Cash, UK and Francesca Mazzia, Italy; two more packages dealing with differential equations are rootSolve and ReacTran.
A book was published about this subject; the examples given here are taken from this book. The other codes can be found in the R-package diffEq.