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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation .
OdeGearControl: Example and Test
Define @(@ X : \B{R} \rightarrow \B{R}^2 @)@ by @[@ \begin{array}{rcl} X_0 (t) & = & - \exp ( - w_0 t ) \\ X_1 (t) & = & \frac{w_0}{w_1 - w_0} [ \exp ( - w_0 t ) - \exp( - w_1 t )] \end{array} @]@ It follows that @(@ X_0 (0) = 1 @)@, @(@ X_1 (0) = 0 @)@ and @[@ \begin{array}{rcl} X_0^{(1)} (t) & = & - w_0 X_0 (t) \\ X_1^{(1)} (t) & = & + w_0 X_0 (t) - w_1 X_1 (t) \end{array} @]@ The example tests OdeGearControl using the relations above: 

# include <cppad/cppad.hpp>
# include <cppad/utility/ode_gear_control.hpp>   // CppAD::OdeGearControl

namespace {
    // --------------------------------------------------------------
    class Fun {
    private:
         CPPAD_TESTVECTOR(double) w;
    public:
        // constructor
        Fun(const CPPAD_TESTVECTOR(double) &w_) : w(w_)
        { }

        // set f = x'(t)
        template <class Scalar>
        void Ode(
            const Scalar                    &t,
            const CPPAD_TESTVECTOR(Scalar) &x,
            CPPAD_TESTVECTOR(Scalar)       &f)
        {   f[0] = - w[0] * x[0];
            f[1] = + w[0] * x[0] - w[1] * x[1];
        }

        void Ode_dep(
            const double                    &t,
            const CPPAD_TESTVECTOR(double) &x,
            CPPAD_TESTVECTOR(double)       &f_x)
        {   using namespace CppAD;

            size_t n  = x.size();
            CPPAD_TESTVECTOR(AD<double>) T(1);
            CPPAD_TESTVECTOR(AD<double>) X(n);
            CPPAD_TESTVECTOR(AD<double>) F(n);

            // set argument values
            T[0] = t;
            size_t i, j;
            for(i = 0; i < n; i++)
                X[i] = x[i];

            // declare independent variables
            Independent(X);

            // compute f(t, x)
            this->Ode(T[0], X, F);

            // define AD function object
            ADFun<double> fun(X, F);

            // compute partial of f w.r.t x
            CPPAD_TESTVECTOR(double) dx(n);
            CPPAD_TESTVECTOR(double) df(n);
            for(j = 0; j < n; j++)
                dx[j] = 0.;
            for(j = 0; j < n; j++)
            {   dx[j] = 1.;
                df = fun.Forward(1, dx);
                for(i = 0; i < n; i++)
                    f_x [i * n + j] = df[i];
                dx[j] = 0.;
            }
        }
    };
}

bool OdeGearControl(void)
{   bool ok = true;     // initial return value
    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

    CPPAD_TESTVECTOR(double) w(2);
    w[0] = 10.;
    w[1] = 1.;
    Fun F(w);

    CPPAD_TESTVECTOR(double) xi(2);
    xi[0] = 1.;
    xi[1] = 0.;

    CPPAD_TESTVECTOR(double) eabs(2);
    eabs[0] = 1e-4;
    eabs[1] = 1e-4;

    // return values
    CPPAD_TESTVECTOR(double) ef(2);
    CPPAD_TESTVECTOR(double) maxabs(2);
    CPPAD_TESTVECTOR(double) xf(2);
    size_t                nstep;

    // input values
    size_t  M   = 5;
    double ti   = 0.;
    double tf   = 1.;
    double smin = 1e-8;
    double smax = 1.;
    double sini = eps99;
    double erel = 0.;

    xf = CppAD::OdeGearControl(F, M,
        ti, tf, xi, smin, smax, sini, eabs, erel, ef, maxabs, nstep);

    double x0 = exp(-w[0]*tf);
    ok &= NearEqual(x0, xf[0], 1e-4, 1e-4);
    ok &= NearEqual(0., ef[0], 1e-4, 1e-4);

    double x1 = w[0] * (exp(-w[0]*tf) - exp(-w[1]*tf))/(w[1] - w[0]);
    ok &= NearEqual(x1, xf[1], 1e-4, 1e-4);
    ok &= NearEqual(0., ef[1], 1e-4, 1e-4);

    return ok;
}
Input File: example/utility/ode_gear_control.cpp