 Implicit differential equation

An implicit ordinary differential equation relates the derivative to the independent and dependent variables without requiring it to be explicitly isolated. For example $y{y}^{\prime }=1$ is meaningful at every pair $(x,y)$ as an equation, but no differentiable solution of a differential equation can pass through $y=0$: its left side would vanish there. Dividing by $y$ gives $y^{\prime }=1/y$ only on the domain $y\ne 0$. Conversely multiplying an explicit equation by a vanishing denominator can introduce points where the original equation was undefined. A solution of the multiplied equation at such a point need not solve the original initial value problem.