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Clairaut's Differential Equation


\begin{displaymath}
y=x{dy\over dx}+f\left({dy\over dx}\right)
\end{displaymath}

or

\begin{displaymath}
y=px+f(p),
\end{displaymath}

where $f$ is a Function of one variable and $p\equiv dy/dx$. The general solution is $y=cx+f(c)$. The singular solution Envelopes are $x=-f'(c)$ and $y=f(c)-cf'(c)$.

See also d'Alembert's Equation


References

Boyer, C. B. A History of Mathematics. New York: Wiley, p. 494, 1968.




© 1996-9 Eric W. Weisstein
1999-05-26