Clairaut equation

An ordinary first-order differential equation not solved with respect to its derivative:

$$y=xy'+f(y'),\label{1}\tag{1}$$

where $f(t)$ is a non-linear function. Equation \eqref{1} is named after A. Clairaut [1] who was the first to point out the difference between the general and the singular solutions of an equation of this form. The Clairaut equation is a particular case of the Lagrange equation.

If $f(t)\in C^1(a,b)$ and $f'(t)\neq0$ when $t\in(a,b)$, then the set of integral curves (cf. Integral curve) of \eqref{1} consists of: a parametrically given curve

$$x=-f'(t),\quad y=-tf'(t)+f(t),\quad a<t<b;\label{2}\tag{2}$$

a one-parameter family of straight lines

$$y=Cx+f(C),\quad C\in(a,b),\label{3}\tag{3}$$

tangent to the curve \eqref{2}; curves consisting of an arbitrary segment of the curve \eqref{2} and the two straight lines of the family \eqref{3} tangent to \eqref{2} at each end of this segment. The family \eqref{3} forms the general solution, while the curve \eqref{2}, which is the envelope of the family \eqref{3}, is the singular solution (see [2]). A family of tangents to a smooth non-linear curve satisfies a Clairaut equation. Therefore, geometric problems in which it is required to determine a curve in terms of a prescribed property of its tangents (common to all points of the curve) leads to a Clairaut equation.

The following first-order partial differential equation is also called a Clairaut equation:

$$z=x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}+f\left(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right);$$

it has the integral

$$x=\alpha x+\beta y+f(\alpha,\beta),$$

where $(\alpha,\beta)$ is an arbitrary point of the domain of definition of the function $f(p,q)$ (see [3]).

References

[1]	A. Clairaut, Histoire Acad. R. Sci. Paris (1734) (1736) pp. 196–215
[2]	V.V. Stepanov, "A course of differential equations" , Moscow (1959) (In Russian)
[3]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen $\mathbf{1^\text{er}}$ Ordnung für eine gesuchte Funktion , Akad. Verlagsgesell. (1944)

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)