We consider the ordinary differential equation with , c > 0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b < 0 then no continuous solutions exist, whereas if a+b > 0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x (0)=a which is such that 0a parts per thousand currency signu(x)a parts per thousand currency signx for all x > 0, and that this solution is strictly increasing and concave.