  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".

General Mathematics