Ordinary Differential Equations/d'Alembert

The d'Alembert's Equation, which is sometimes called the Lagrange equation was solved by John Bernoulli before 1694, and d'Alembert studied its singular solutions in a 1748 publication. It is essentially an equation of the form

$y=xf(y')+g(y')$

Where $f$ and $g$ are functions of $y'$.

Take the derivative

$y'=f(y')+(xf'(y')+g'(y'))y''$

Now write this equation as

${\frac {dx}{dy'}}-{\frac {f'(y')}{y'-f(y')}}x={\frac {g'(y')}{y'-f'(y')}}$

Then it is a linear equation with dependent variable x and independent variable <amth>y'</math>.