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:


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