form an ordinary differential equation from the curve v=A/r + B where A and B are constant and then identify it
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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.

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