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+Just to clarify here, I am pretty sure both are possible (I use AD on an ODE solver all the time), see https://docs.sciml.ai/DiffEqDocs/v5.3/analysis/sensitivity.html
+https://arxiv.org/abs/1812.01892
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+AD is quite useful here because you can get the sensitivity of non-analytic things easily. E.g. sensitivity of gravity assist periapsis wrt control. Also using AD on ODE solvers saves a lot of time because you don't need to write out the equations to propagate the Jacobian, they are implicitly created with the dual numbers.
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+As something to watch out for when doing this, you need to be really careful about adaptive timestepping. A timestep which is valid for the original problem may not be valid for the problem with AD - the timestepping algorithm needs to take this into account.