Clairaut's Equation

The equation y = px + f(p) is known as Clairaut's equation.
Differentiating w.r.t. x we get,
dy/dx = p + x(dp/dx) + f'(p) dp/dx
p = p + x[dp/dx] + f'(p) dp/dx 
or
0 = x dp/dx + f'(p) dp/dx


[x + f'(p)]dp/dx = 0 or dp/dx = 0 or p = a (constant)

Putting value of p in (1) we have
y = ax + f(a)
which is the required solution.

In a Clairaut's equation, on replacing p by a constant we get the solution of the equation.



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