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Clairaut's Differential Equation

y=x(dy)/(dx)+f((dy)/(dx))
(1)

or

y=px+f(p),
(2)

where f is a function of one variable and p=dy/dx. The general solution is

y=cx+f(c).
(3)

The singular solution envelopes are x=-f^'(c) and y=f(c)-cf^'(c).

A partial differential equation known as Clairaut's equation is given by

u=xu_x+yu_y+f(u_x,u_y)
(4)

(Iyanaga and Kawada 1980, p. 1446; Zwillinger 1997, p. 132).

See also

Clairaut's Difference Equation, d'Alembert's Equation

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References

Boyer, C. B. A History of Mathematics. New York: Wiley, p. 494, 1968.Ford, L. R. Differential Equations. New York: McGraw-Hill, p. 16, 1955.Ince, E. L. Ordinary Differential Equations. New York: Dover, pp. 39-40, 1956.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1446, 1980.Zwillinger, D. "Clairaut's Equation." §II.A.38 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 120 and 158-160, 1997.

Referenced on Wolfram|Alpha

Clairaut's Differential Equation

Cite this as:

Weisstein, Eric W. "Clairaut's Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClairautsDifferentialEquation.html

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