Clairauts Equation:

An equation of the form

Y = px + ƒ(p)                                          (1)

is called as Clairaut's equation.

To calculate it, differentiate (1) w.r.t. x so that we obtain

p = p + x(dp/dx) + ƒ'(p) (dp/dx)

i.e. [x + ƒ'(p)] (dp/dx) = 0.

Therefore we have (dp/dx) = 0 => p = c, a constant             (2)

Or, x + ƒ' (p) = 0.

Removing p between (1) and (2), we get
 
Y = cx + ƒ(c),                                          (3)

which is the needed solution of (1).

If we remove p between (1) and (3), we obtain a solution which does not have any arbitrary constant. Such a solution is known as the singular solution of (1).

Working Rule: The solution of y = px + ƒ(p) is provided by replacing p by c.