Clairaut's Differential Equation

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


 * $y = x y' + \map f {y'}$

Its general solution is:


 * $y = C + \map f C$

where $C$ is a constant.

Proof
We have:
 * $y = x y' + \map f {y'}$

Differentiating the equation we have:

Proof for General Solution
The first solution is:

By substituting into the original equation, we obtain:

Hence the result:


 * $y = C_1 x + \map f {C_1}$