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

Then it is a linear equation with dependent variable x and independent variable y'.