Implicit factorization can reveal nonunique initial solutions ($(2\sin y-x)y^{\prime }=\tan y,y(0)=0$)

The displayed implicit ordinary differential equation has a singular initial point at the origin. On a nonconstant regular branch, treating $x$ as a function of $y$ gives $dx/dy+x\cot y=2\cos y$. The integrating factor $\sin y$ yields the first integral of an ordinary differential equationOne can verify this identity without dividing by $\sin y$: its derivative along a solution is $\sin y-(2\sin y-x)\cos y{y}^{\prime }=0$.

The initial point forces $C=0$, so $\sin y(x-\sin y)=0$. Near the origin this gives the two differentiable solutions $y=0$ and $y=\arcsin x$, with initial slopes zero and one. A division by $\sin y$ would lose the first solution. A piecewise switch between the two at zero would fail differentiability because their one-sided slopes differ.

There is no conflict with the Picard-Lindelöf theorem: solving explicitly for $y^{\prime }$ requires division by $2\sin y-x$, which vanishes at the initial point. A regular implicit equation or a local uniqueness theorem must be checked before claiming a unique solution through a singular point.