Clairaut's equation
Encyclopedia
In mathematics
, a Clairaut's equation is a differential equation
of the form
To solve such an equation, we differentiate with respect to x, yielding
so
Hence, either
or
In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, we have the family of functions given by
the so-called general solution of Clairaut's equation.
The latter case,
defines only one solution y(x), the so-called singular solution
, whose graph is the envelope
of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as (x(p), y(p)), where p represents dy/dx.
This equation was named after Alexis Clairaut, who introduced it in 1734.
A first-order partial differential equation
is also known as Clairaut's equation or Clairaut equation:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Clairaut's equation is a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
of the form
To solve such an equation, we differentiate with respect to x, yielding
so
Hence, either
or
In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, we have the family of functions given by
the so-called general solution of Clairaut's equation.
The latter case,
defines only one solution y(x), the so-called singular solution
Singular solution
A singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...
, whose graph is the envelope
Envelope (mathematics)
In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...
of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as (x(p), y(p)), where p represents dy/dx.
This equation was named after Alexis Clairaut, who introduced it in 1734.
A first-order partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
is also known as Clairaut's equation or Clairaut equation: