  Identifying computational tasks suitable for (future) quantum computers is an
active field of research. Here we explore utilizing quantum computers for the
purpose of solving differential equations. We consider two approaches: (i)
basis encoding and fixed-point arithmetic on a digital quantum computer, and
(ii) representing and solving high-order Runge-Kutta methods as optimization
problems on quantum annealers. As realizations applied to two-dimensional
linear ordinary differential equations, we devise and simulate corresponding
digital quantum circuits, and implement and run a 6$^{\mathrm{th}}$ order
Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good
agreement with the reference solution. We find that the quantum annealing
approach exhibits the largest potential for high-order implicit integration
methods. As promising future scenario, the digital arithmetic method could be
employed as an "oracle" within quantum search algorithms for inverse problems.
