# Ordinary Differential Equations/d'Alembert

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'.