There are surely many aspects to this question. In particular, the inverse problems are often ill-posed problems. For example, when you reverse time in the heat equation, you have an ill-posed problem because it is extraordinarily sensitive to initial conditions.
See for exemple well-posed problem
Problems that are not well-posed in the sense of Hadamard are termed
ill-posed. Inverse problems are often ill-posed. For example, the
inverse heat equation, deducing a previous distribution of temperature
from final data, is not well-posed in that the solution is highly
sensitive to changes in the final data.
For the damped harmonic oscillator, the time reversal leads to an unstable equation (dissipative force are not reversible) which is also very sensitive to the initial conditions. For example, if you start from rest (0,0) the solution x(t) = 0 but if you start from $(ε, 0)$ the solution will be exponentially divergent.