# Clairaut's Differential Equation

(Redirected from Clairaut's 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:

$y = C x + \map f C$

where $C$ is a constant.

## Proof

We have:

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

Differentiating the equation with respect to $x$ we have:

 $\displaystyle y'$ $=$ $\displaystyle y' + x y'' + y'' \map {f'} {y'}$ $\displaystyle \leadsto \ \$ $\displaystyle 0$ $=$ $\displaystyle \map {y''} {x + \map {f'} {y'} }$

### Proof for General Solution

The first solution is:

 $\displaystyle y''$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle y'$ $=$ $\displaystyle C_1$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle C_1 x + C_2$

By substituting into the original equation, we obtain:

 $\displaystyle C_1 x + C_2$ $=$ $\displaystyle x C_1 + \map f {C_1}$ $\displaystyle \leadsto \ \$ $\displaystyle C_2$ $=$ $\displaystyle \map f {C_1}$

Hence the result:

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

$\blacksquare$

## Source of Name

This entry was named for Alexis Claude Clairaut.