# Are there shortcuts for numerically approximating systems of ordinary differential equations when autonomous?

Existing algorithms for solving ODEs handle functions $\frac{dy}{dt} = f(y, t)$, where $y \in \mathbb R^n$. But in many physical systems, the differential equation is autonomous, so $\frac{dy}{dt} = f(y)$, $y \in \mathbb R^n$, with the $t$ left out. With this simplifying assumption, what improvements can be seen in existing numerical methods? For example, if $n=1$, the problem turns into $t = \int \frac{dy}{f(y)}$ and we turn to a wholly different class of algorithms for integrating one-dimensional integrals. For $n>1$, the maximum possible improvement is reducing the dimension of $y$ by 1, because the time-dependent case can be simulated by appending $t$ to $y$, changing the domain of $y$ from $\mathbb R^n$ to $\mathbb R^{n+1}$.

I would say one significant improvement is that in the scope of time-stepping approaches, where you propagate $$y_n \rightarrow y_{n+1}=\mathcal{U}(y_n)$$ using a solution map $$\mathcal{U}$$, you can determine the propagator (or at least parts of it) once and then re-use it at every time step.
For example, in the linear case you would have $$\partial_t y = A y$$, where $$A$$ is a matrix. The solution operator $$\mathcal{U}(y) = \exp(A \Delta t)y$$ consists mainly of a matrix exponential. For autonomous systems, this costly matrix exponential evaluation is required only once for the complete propagation -- in contrast to a time-dependent system, where you have to perform this evaluation at every time step.