Editing Envelope (mathematics) (section)

=== Ordinary differential equations ===
Envelopes are connected to the study of [[ordinary differential equation]]s (ODEs), and in particular [[singular solution]]s of ODEs.<ref>{{Citation | last1=Forsyth | first1=Andrew Russell | title=Theory of differential equations | publisher=[[Dover Publications]] | location=New York | series=Six volumes bound as three | mr=0123757 | year=1959}}, §§100-106.</ref> Consider, for example, the one-parameter family of tangent lines to the parabola ''y'' = ''x''<sup>2</sup>. These are given by the generating family {{nowrap|1=''F''(''t'',(''x'',''y'')) = ''t''<sup>2</sup> – 2''tx'' + ''y''}}. The zero level set {{nowrap|1=''F''(''t''<sub>0</sub>,(''x'',''y'')) = 0}} gives the equation of the tangent line to the parabola at the point (''t''<sub>0</sub>,''t''<sub>0</sub><sup>2</sup>). The equation {{nowrap|1=''t''<sup>2</sup> – 2''tx'' + ''y'' = 0}} can always be solved for ''y'' as a function of ''x'' and so, consider
:<math> t^2 - 2tx + y(x) = 0. \ </math>
Substituting
:<math> t = \left(\frac{dy}{dx}\right)/2 </math>
gives the ODE
:<math> \left(\frac{dy}{dx}\right)^2 \!\! - 4x\frac{dy}{dx} + 4y = 0. </math>
Not surprisingly ''y''&nbsp;=&nbsp;2''tx''&nbsp;&minus;&nbsp;''t''<sup>2</sup> are all solutions to this ODE. However, the envelope of this one-parameter family of lines, which is the parabola ''y''&nbsp;=&nbsp;''x''<sup>2</sup>, is also a solution to this ODE. Another famous example is [[Clairaut's equation]].