Families of curves orthogonal to the lines $y=mx-2m-m^3$

Abstract

The family of lines $y=mx-2m-m^3$, are well known to be normal to the parabola $y^2=4x$. However, this family of lines is normal to a family of curves of which this parabola is just one member. Here, by solving an interesting first order and third degree ODE, we bring out these curves. The resulting one set of curves are "parabola-like" but non-standard ones and the other family is not even "parabola like".