Clairaut's equation
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 straight line 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 has been 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:
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Examples
-
Solutions to Clairaut's equation where f(t)=t^2.png
Solutions to Clairaut's equation where
-
Solutions to Clairaut's equation where f(t)=t^3.png
External links
- Lua error in package.lua at line 80: module 'strict' not found.. At Gallica: the paper of Clairaut introducing the equation named after him.
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