d’Alembert’s equation

The first differential equation

$$y=\phi (\frac{dy}{dx})\cdot x+\psi (\frac{dy}{dx})$$

is called d’Alembert’s differential equation; here $\phi$ and $\psi$ some known differentiable real functions.

If we denote  $\frac{dy}{dx}:=p$, the equation is

$$y=\phi (p)\cdot x+\psi (p).$$

We take $p$ as a new variable and derive the equation with respect to $p$, getting

$$p-\phi (p)=[x{\phi }^{\prime }(p)+{\psi }^{\prime }(p)]\frac{dp}{dx}.$$

If the equation  $p-\phi (p)=0$  has the roots  $p=p_1$, $p_2$, …, $p_k$, then we have  $\frac{d{p}_{\nu }}{dx}=0$  for all $\nu$’s, and therefore there are the special solutions

$$y=p_{\nu }x+\psi (p_{\nu })\mathit{\hspace{1em}}(\nu =1,2,\mathrm{\dots },k)$$

for the original equation.  If  $\phi (p)\not\equiv p$, then the derived equation may be written as

$$\frac{dx}{dp}=\frac{{\phi }^{\prime }(p)}{p-\phi (p)}x+\frac{{\psi }^{\prime }(p)}{p-\phi (p)},$$

which linear differential equation has the solution  $x=x(p,C)$  with the integration constant $C$.  Thus we get the general solution of d’Alembert’s equation as a parametric

$$\{\begin{array}{cc}x=x(p,C),\hfill & \\ y=\phi (p)x(p,C)+\psi (p)\hfill & \end{array}$$

of the integral curves.