to the parabola at the point (t0,t02). The equation can always be solved for y as a function of x and so, consider Substituting gives the ODE Not surprisingly y = 2tx − t2 are all solutions to this ODE. However, the envelope of this one-parameter family of lines, which is the parabola y = x2, is also a solution to this ODE. Another famous example is Clairaut's equation. Partial differential equations Envelopes can be used to construct more complicated solutions of first order partial differential equations (PDEs) from simpler ones. Let F(x,u,Du) = 0 be a first order PDE, where x is a variable with values in an open set Ω ⊂ Rn, u is an unknown real-valued function, Du is the gradient of u, and F is a continuously differentiable function that is regular in Du. Suppose that u(x;a) is an m-parameter family of solutions: that is, for each fixed a ∈ A ⊂ Rm, u(x;a) is a solution of the differential equation. A new solution of the differential equation can be constructed by first solving (if possible) for a = φ(x) as a function of x. The envelope of the family of functions {u(·,a)}a∈A is defined by and also solves the differential equation (provided that it exists as a continuously differentiable function). Geometrically, the graph of v(x) is everywhere tangent to the graph of some member of the family u(x;a). Since the differential equation is first order, it only puts a condition on the tangent plane to the graph, so that any function everywhere tangent to a solution must also be a solution. The same idea underlies the solution of a first order equation as an integral of the Monge cone. The Monge cone is a cone field in the Rn+1 of the (x,u) variables cut out by the envelope of the tangent spaces to the first order PDE at each point. A solution of the PDE is then an envelope of the cone field. In Riemannian geometry, if a smooth family of geodesics through a point P in a Riemannian manifold has an envelope, then P has a conjugate point where any geodesic of the family intersects the envelope. The same is true more generally in the calculus of variations: if a family of extremals to a functional through a given point P has an envelope, then a point where an extremal intersects the envelope is