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

$y=xf(y^{\prime })+g(y^{\prime })$

Where $f$ and $g$ are functions of $y^{\prime }$.

Take the derivative

$y^{\prime }=f(y^{\prime })+(x{f}^{\prime }(y^{\prime })+g^{\prime }(y^{\prime }))y^{″}$

Now write this equation as

$\frac{dx}{d{y}^{\prime }}-\frac{f^{\prime }(y^{\prime })}{y^{\prime }-f(y^{\prime })}x=\frac{g^{\prime }(y^{\prime })}{y^{\prime }-f^{\prime }(y^{\prime })}$

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