Clairaut's Differential Equation

Theorem
Clairaut's differential equation is a first order ordinary differential equation which can be put into the form:


 * $$y = x y' + f \left({y'}\right)$$

Its general solution is:


 * $$y = C + f \left({C}\right)$$

where $$C$$ is a constant.

Proof
We have:
 * $$y = x y' + f \left({y'}\right)$$

Differentiating the equation we have:

$$ $$

Proof for general solution
The first solution is:

$$ $$ $$

By substituting into the original equation, we obtain:

$$ $$

Hence the result:


 * $$y = C_1 x + f \left({C_1}\right)$$