Necessary and sufficient conditions for a curve to be the gradient range of a $C^1$-smooth function

We find some necessary and sufficient conditions for a plane curve to be the gradient range of a $C^1$-smooth function of two variables. As one of the consequences we give the necessary and sufficient conditions on a continuous function $\varphi$ under which the differential equation $\dfrac{\partial v}{\partial t}=\varphi\biggl(\dfrac{\partial v}{\partial x}\biggr)$ has nontrivial $C^1$-smooth solutions.