v=A/r+B.
We need an ODE which doesn’t involve A and B. Because there are two constants, we can guess that the ODE is second order, so that the two constants arise from integration.
Differentiate:
v'=-A/r² and differentiating again, v''=2A/r³.
Therefore, v'/v''=-r/2, 2v'=-rv'', rv''+2v'=0 is the required second order ODE.