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We consider the ordinary differential equation $x^2 u'' = axu'+bu-c(u'-1)^2,x\in (0,x_0)$, with $a\in\mathbb{R}, b\in\mathbb{R}$, $c>0$ and the singularinitial condition $u(0)=0$, which in financial economics describes optimaldisposal of an asset in a market with liquidity effects. It is shown in thepaper that if $a+b < 0$ then no solutions exist, whereas if $a+b\ge0$ thenthere are infinitely many 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 $x_0=\infty$ which is such that $0 \le u(x)\lex$ for all $x>0$, and that this solution is strictly increasing and concave.

A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics