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 y'.