A kind of problem that turns up quite often in physics and elsewhere is to find the solution of an ordinary differential equation, given some initial conditions. That is, you have a system with some state $x$ represented as a vector of real numbers, you have an initial state $x_0$, and you have a rule describing the evolution of the state:
$$
\frac{dx}{dt} = F(x,t)
$$
And your goal is to find $x(t)$. This standard problem has some standard solution techniques, some quite advanced - Runge-Kutta methods, symplectic methods, Hermite integrators. A few are implemented in scipy. But it sometimes happens that solving this problem is only part of a more complicated process, say of fitting, where it would be nice to have the derivatives of the solution with respect to the various initial conditions. It turns out this isn't too hard to work out, usually.

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