GNU Scientific Library Reference Manual  Third Edition (v1.12) by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) 
25.1 Defining the ODE System
The routines solve the general ndimensional firstorder system,
dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
for i = 1, \dots, n. The stepping functions rely on the vector
of derivatives f_i and the Jacobian matrix,
J_{ij} = df_i(t,y(t)) / dy_j.
A system of equations is defined using the gsl_odeiv_system
datatype.
 Data Type: gsl_odeiv_system
 This data type defines a general ODE system with arbitrary parameters.
int (* function) (double t, const double y[], double dydt[], void * params)

This function should store the vector elements
f_i(t,y,params) in the array dydt,
for arguments (t,y) and parameters params.
The function should return
GSL_SUCCESS
if the calculation was completed successfully. Any other return value indicates an error. int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);

This function should store the vector of derivative elements
df_i(t,y,params)/dt in the array dfdt and the
Jacobian matrix
J_{ij} in the array
dfdy, regarded as a rowordered matrix
J(i,j) = dfdy[i * dimension + j]
wheredimension
is the dimension of the system. The function should returnGSL_SUCCESS
if the calculation was completed successfully. Any other return value indicates an error. Some of the simpler solver algorithms do not make use of the Jacobian matrix, so it is not always strictly necessary to provide it (thejacobian
element of the struct can be replaced by a null pointer for those algorithms). However, it is useful to provide the Jacobian to allow the solver algorithms to be interchangedthe best algorithms make use of the Jacobian. size_t dimension;
 This is the dimension of the system of equations.
void * params
 This is a pointer to the arbitrary parameters of the system.
ISBN 0954612078  GNU Scientific Library Reference Manual  Third Edition (v1.12)  See the print edition 