mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

, Clairaut's equation (or the Clairaut equation) is a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

of the form :

y(x)=x\frac+f\left(\frac\right)

where ''f'' is

continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

. It is a particular case of the Lagrange differential equation. It is named after the French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Alexis Clairaut Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had ou ...

, who introduced it in 1734.

Definition

To solve Clairaut's equation, one differentiates with respect to ''x'', yielding :

\frac=\frac+x\frac+f'\left(\frac\right)\frac,

so :

frac = 0.

Hence, either :

\frac = 0

or :

x+f'\left(\frac\right) = 0.

In the former case, ''C'' = ''dy''/''dx'' for some constant ''C''. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by :

y(x)=Cx+f(C),\,

the so-called ''general solution'' of Clairaut's equation. The latter case, :

x+f'\left(\frac\right) = 0,

defines only one solution ''y''(''x''), the so-called '' singular solution'', whose graph is the

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...

of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as (''x''(''p''), ''y''(''p'')), where ''p'' = ''dy''/''dx''. The parametric description of the singular solution has the form :

x(t)= -f'(t),\,

y(t)= f(t) - tf'(t),\,

where t is a parameter.

Examples

The following curves represent the solutions to two Clairaut's equations: Image:Solutions to Clairaut's equation where f(t)=t^2.png, Image:Solutions to Clairaut's equation where f(t)=t^3.png, In each case, the general solutions are depicted in black while the singular solution is in violet.

Extension

By extension, a first-order

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

of the form :

\displaystyle u=xu_x+yu_y+f(u_x,u_y)

is also known as Clairaut's equation..

Notes

References

*. *. *{{springer , title = Clairaut equation , id = C/c022350 , last = Rozov , first = N. Kh. . Ordinary differential equations

Definition

Examples

Extension

See also

Notes

References