Define $y : \B{R} \times \B{R} \rightarrow \B{R}^n$ by $$y_j (x, t) = x t^{j+1}$$ It follows that the derivative of $y(t)$ satisfies the runge_kutta_4 ODE equation where $y(0) = 0$ and $f(t, y)$ is given by $$f(t , y)_j = \partial_t y_j (x, t) = \left\{ \begin{array}{ll} x & {\; \rm if \;} j = 0 \\ (j+1) y_{j-1} (x, t) & {\; \rm otherwise } \end{array} \right.$$ It also follows that $$\partial_x y_j (x, t) = t^{j+1}$$
Source Code  from pycppad import * def pycppad_test_runge_kutta_4_ad() : def fun(t , y) : n = y.size f = ad( numpy.zeros(n) ) f[0] = a_x[0] index = numpy.array( range(n-1) ) + 1 f[index] = (index + 1) * y[index-1] return f n = 5 # size of y(t) (order of method plus 1) ti = 0. # initial time dt = 2. # a very large time step size (method is exact) # initial value for y(t); i.e., y(0) a_yi = ad( numpy.zeros(n) ) # declare a_x to be the independent variable vector x = numpy.array( [.5] ) a_x = independent( numpy.array( x ) ) # take one 4-th order Runge-Kutta integration step of size dt a_yf = runge_kutta_4(fun, ti, a_yi, dt) # define the AD function g : x -> yf g = adfun(a_x, a_yf) # compute the derivative of g w.r.t x at x equals .5 dg = g.jacobian(x) # check the result is as expected t_jp = 1 # t^0 for j in range(n-1) : t_jp = t_jp * dt # t^(j+1) at t = dt assert abs( a_yf[j] - x[0]*t_jp ) < 1e-10 # check yf[j] = x*t^(j+1_ assert abs( dg[j,0] - t_jp ) < 1e-10 # check dg[j] = t^(j+1)